AWK, 21 bytes

1,gsub(/\B[aeiou]/,X)

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Uiua, 16 bytes

⍜↘(▽¬∊,"aeiou")1

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⍜↘(▽¬∊,"aeiou")1
⍜↘(           )1  # apply (...) to the input except the first letter
   ▽¬∊,"aeiou"    # remove vowels

TI-Basic, 58 bytes

Input Str1
sub(Str1,1,1
For(I,2,length(Str1
If not(inString("AEIOU",sub(Str1,I,1
Ans+sub(Str1,I,1
End
Ans

Takes input as all capital letters (which is hopefully allowed).

Vyxal s, 5 bytes

ḣkvF"

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Port of the 05Ab1e answer but one less byte because flags are cool

Arturo, 29 bytes

$[s][replace s{/\B[aeiou]}""]

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Factor, 31 bytes

[ 1 cut "aeiou"without append ]

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Explanation

It's a quotation (anonymous function) that takes one string as input and leaves one string as output. Assuming "potato" is on top of the data stack when this quotation is called...

• 1 Push one to the data stack. Stack: "potato" 1
• cut Cut a sequence into two pieces at a given index. Stack: "p" "otato"
• "aeiou" Push a string to the data stack. Stack: "p" "otato" "aeiou"
• without Remove elements from NOS (next on stack) that are in TOS (top of stack). Stack: "p" "tt"
• append Concatenate two strings. Stack: "ptt"

05AB1E, 6 bytes

ćsžMм«

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Explanation

       Input (e.g.). potato
ć      Head extract. otato, p
 s     Swap        . p, otato
  žM   Vowels      . p, otato, aeiou
    м  Remove      . p, tt
     « Concatenate . ptt

Implicit output    .

Excel, 87 bytes

=LET(q,SEQUENCE(LEN(A1)),x,MID(A1,q,1),CONCAT(IF(ISERROR(FIND(x,"aeiou"))+(q=1),x,"")))

Hopefully this link works. Try it here

naz, 102 bytes

2x1v2a6m8m1a2x2v4a2x3v4a2x4v6a2x5v6a2x6v1x1f1r3x1v2e3x2v1e3x3v1e3x4v1e3x5v1e3x6v1e1o1f0x1x2f0a0x1r1o1f

Works for any null-terminated input string.

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Explanation (with 0x instructions removed)

2x1v                                           # Set variable 1 equal to 0
2a6m8m1a2x2v                                   # Set variable 2 equal to 97 ("a")
4a2x3v                                         # Set variable 3 equal to 101 ("e")
4a2x4v                                         # Set variable 4 equal to 105 ("i")
6a2x5v                                         # Set variable 5 equal to 111 ("o")
6a2x6v                                         # Set variable 6 equal to 117 ("u")
1x1f                                           # Function 1
    1r                                         # Read a byte of input
      3x1v2e                                   # Goto function 2 if it equals variable 1
            3x2v1e3x3v1e3x4v1e3x5v1e3x6v1e     # Jump back to the start of the function
                                               # if it equals variable 2, 3, 4, 5, or 6
                                          1o1f # Otherwise, output it, then call the
                                               # function again
1x2f                                           # Function 2
    0a                                         # Add 0 to the register
1r1o                                           # Output the first byte of input
1f                                             # Call function 1

Japt v2.0a0, 7 bytes

Î+UÅr\v

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Stax, 5 bytes

ö¶╫♂.

Run and debug it

Brachylog, 7 bytes

h|b;Ḅ∋ᵛ

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Generator.

 |         The output is
h          the first element of the input,
 |         or
     ∋ᵛ    some shared element of
 |b        the input without its first element
   ;Ḅ      and the lowercase consonants.

JavaScript (V8), 42 bytes

s=>s[0]+s.substr(1).replace(/[aeiou]/g,'')

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Ruby -p, 22 19 bytes

gsub /\B[aeiou]/,''

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Thanks to Command Master for -3 bytes.

Pyth, 12 bytes

+hQ-tQ"aeiou

Explanation

+hQ-tQ"aeiou
    tQ        : Everything except first element of evaluated input
   -  "aeiou  : Remove all occurrences of a, e, i, o and u from the string
+hQ           : Prepend first element of evaluated input

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Jelly, 6 bytes

Ḣ;ḟØẹ$

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How?

Ḣ;ḟØẹ$ - Link: list of characters, W
Ḣ      - head & yield
     $ - last two links as a monad - i.e. f(rest of W)
  ḟ    -   filter discard:
   Øẹ  -     lower-case vowels
 ;     - (head of W) concatenate (f(rest of W))

CJam, 15 bytes

l_0=\1>"aeiou"-

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It's 15 bytes because there is no better way to push all those vowels.

Explained

l_0=\1>"aeiou"-
l_                  e# Take the input and duplicate it: [input, input]
  0=                e# From the input, take the first character: [input, input[0]]
    \               e# Swap the top two items: [input[0], input]
     1>             e# Drop the first character: [input[0], input[1:]]
       "aeiou"      e# Push the string "aeiou": [input[0], input[1:], "aeiou"]
              -     e# Remove all occurances of the above string from the input[1:] and print.

Keg -ir, 34 16 bytes

Huge saving thanks to Lyxal.

,(:⅍`aeiou`-[,|_

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PowerShell, 33 30 bytes

-3 bytes thanks to @CommandMaster

$args-split"(.)[aeiou]"-join''

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PHP, 41 bytes

<?=preg_replace("/\\B[aeiou]/","",$argn);

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I wanted to find an original solution but it ends up like everyone's regex.. Deception is my middle name

APL (NARS2000 0.5.13.0), 20 unicode char (so 20 byte or 40 byte?)

(↑,((~∘'aeiou')1∘↓))

Sample output:

      (↑,((~∘'aeiou')1∘↓)) 'i'
i
      (↑,((~∘'aeiou')1∘↓)) 'ate'
at
      (↑,((~∘'aeiou')1∘↓)) 'potato'
ptt
      (↑,((~∘'aeiou')1∘↓)) 'ksboejqfgcrauxpwivtzndlhmy' 
ksbjqfgcrxpwvtzndlhmy
      (↑,((~∘'aeiou')1∘↓)) ''

Thanks to the tacit feature added lately.

Since APL code chars are unicode chars, I am not sure if one char should be counted as two byte or not.

JavaScript (ES6),  27  26 bytes

Saved 1 byte thanks to @Neil

Returns a list of characters.

s=>s.match(/^.|[^aeiou]/g)

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Perl 5 -p, 18 bytes

s/(?<!^)[aeiou]//g

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Ruby, 31 bytes

->x{x[0]+x[1..].tr('aeiou','')}

Needs to be run on Ruby 2.6 or above, since it uses an endless range.

(TIO is on 2.5.5, so it doesn't work there.)

Charcoal, 12 bytes

ΦＳ¬∧κ№aeiouι

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 Ｓ              Input string
Φ ¬             Exclude characters where
     №aeiouι    Character is a vowel
   ∧κ           And index is not zero
                Implicitly print

Python 3, ⁵⁷ 55 bytes

Saved 2 bytes thanks to ovs!!!

lambda s:s[0]+''.join(c for c in s[1:]if{c}-{*'aeiou'})

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Python 2, 43 bytes

Another 43-byte solution in Python.

lambda s:s[0]+s[1:].translate(None,'aeiou')

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Python 3, 61 bytes

lambda s:s[0]+''.join(filter(lambda x:x not in'aeiou',s[1:]))

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Haskell, 34 bytes

f(a:s)=a:[x|x<-s,notElem x"aeiou"]

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K (oK), 20 bytes

{(*x),(1_x)^"aoeiu"}

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C (gcc), 57 56 52 bytes

Time to roll out some magic numbers!

Thanks @Arnauld for even better magic numbers!

f(char*a,char*b){for(*b++=*a;*b=*++a;)b+=4373%*a&1;}

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sed, 14 bytes

s/\B[aeiou]//g

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Bash + Core utilities, 19 bytes

sed s/\\B[aeiou]//g

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Retina, 15 13 8 bytes

0T1,`v`_

Transliterates lowercase vowels into nothing (_) in the 0th in 0-indexing match of the implicit regex that matches the entire string, after index 1 (1,).

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APL (Dyalog), 14 bytes

⊃,'aieou'~⍨1∘↓

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Erlang (escript), 26 bytes

f([H|T])->[H]++T--"aeiou".

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Explanation

f([H|T])->       % Match the head & tail of the input string.
[H]              % Wrap the head in a list,
   ++T           % Append the tail
      --"aeiou". % with all vowels removed.

Python 3, 44 43 bytes

lambda a,*s:[a]+[c[c in"aeiou":]for c in s]

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Input: Characters of the word. E.g f("a", "t", "e")
Output: A list of characters.

SNOBOL4 (CSNOBOL4), 66 bytes

 INPUT LEN(1) . F REM . I
V I ANY('aeiou') =:S(V)
 OUTPUT =F I
END

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Red, 33 bytes

func[s][trim/with next s"aeiou"s]

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J, 15 bytes

{.,'aeiou'-.~}.

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Straightforward:

• {. head...
• , catted with...
• 'aeiou'-.~ vowels, set minused from...
• }. tail