Ruby -ripaddr, 14 bytes

This is kind of cheating, but the Ruby standard library has an IPAddr class that you can construct ranges from.

->a,b{[*a..b]}

Attempt This Online!

Ruby -ripaddr, 47 bytes

If you want to insist on string input and array-of-strings output, it’s a bit longer.

->a,b{((i=IPAddr).new(a)..i.new(b)).map &:to_s}

Attempt This Online!

JavaScript (ES6), 104 bytes

g=s=>s.replace(/(\d+)\.256/,(_,n)=>++n+'.0')
f=(a,b)=>a==b?a:a+`
`+f(g(g(g(g(a+'.256').slice(0,-2)))),b)

This solution replaces the pattern n.256 with n+1.0, and it calls itself recursively until the two parameters are equal.

It appends '.256' to the initial input to get the ball rolling. slice(0,-2) is then used to remove the trailing '.0'.

Examples:

g=s=>s.replace(/(\d+)\.256/,(_,n)=>++n+'.0')
f=(a,b)=>a==b?a:a+`
`+f(g(g(g(g(a+'.256').slice(0,-2)))),b)

console.log('192.168.0.1 ... 192.168.0.4');
console.log(f('192.168.0.1', '192.168.0.4'));

console.log('123.255.255.0 ... 124.0.3.0');
console.log(f('123.255.255.0', '124.0.3.0'));

console.log('192.1.1.1 ... 192.1.1.1 ... kudos');
console.log(f('192.1.1.1', '192.1.1.1'));

Java (OpenJDK 8), 339 314 282 bytes

Golfed:

long f(String i)throws Exception{return java.nio.ByteBuffer.wrap(java.net.InetAddress.getByName(i).getAddress()).getInt()&(1L<<32)-1;}void g(String[] a)throws Exception{for(long l=f(a[0]),m=255;l<=f(a[1]);)System.out.println(((l>>24)&m)+"."+((l>>16)&m)+"."+((l>>8)&m)+"."+(l++&m));}

Ungolfed:

long ipToLong(String ip) throws Exception {
    return java.nio.ByteBuffer.wrap(java.net.InetAddress.getByName(ip).getAddress()).getInt() & (1L << 32) - 1;
}

void printRange(String[] ips) throws Exception {
    for (long ip = ipToLong(ips[0]), m = 255; ip <= ipToLong(ips[1]);)
        System.out.println(((ip >> 24) & m) + "." + ((ip >> 16) & m) + "." + ((ip >> 8) & m) + "." + (ip++ & m));
}

Try it online!

Jelly, 18 bytes

ṣ”.V€ḅ⁹µ€r/b⁹j”.$€

Try it online!

The output appears to by a mush of digits and decimals, but is stored internally as a list of strings. Append a Y to the end (+1 byte) to join the strings by newlines.

How it Works

ṣ”.V€ḅ⁹µ€r/b⁹j”.$€ - main link, input is a list of addresses which are strings
       µ€          - for each address string,
ṣ”.                  - split on periods
   V€                - eval each element in the list (convert strings to numbers)
     ḅ⁹              - transform from base 256 to integer (⁹ is 256)
         r/        - take the inclusive range
           b⁹      - convert each element in the range from integer to base 256
             j”.$€ - join each address with periods.

Batch, 623 bytes

@echo off
set s=%1.%2
call:c %s:.= %
exit/b
:c
if %1==%5 goto d
call:d %1 %2 %3 %4 %1 255 255 255
set/al=%1+1,u=%5-1
for /l %%i in (%l%,1,%u%)do call:d %%i 0 0 0 %%i 255 255 255
call:d %5 0 0 0 %5 %6 %7 %8
exit/b
:d
if %2==%6 goto e
call:e %1 %2 %3 %4 %1 %2 255 255
set/al=%2+1,u=%6-1
for /l %%j in (%l%,1,%u%)do call:e %1 %%j 0 0 %5 %%j 255 255
call:e %5 %6 0 0 %5 %6 %7 %8
exit/b
:e
if %3==%7 goto f
call:f %1 %2 %3 %4 %1 %2 %3 255
set/al=%3+1,u=%7-1
for /l %%k in (%l%,1,%u%)do call:e %1 %2 %%k 0 %5 %6 %%k 255
call:e %5 %6 %7 0 %5 %6 %7 %8
exit/b
:f
for /l %%l in (%4,1,%8)do echo %1.%2.%3.%%l

Unfortunately Batch's 32-bit arithmetic can't print all IP addresses, so I have to split it up into octets.

Pyth, 22

mj\.jd256}FmivMcd\.256

Try it online.

               cd\.       # split string by "."
             vM           # eval list (convert strings to integers)
            i      256    # convert list of base256 digits to integer
           m          Q   # map the above over implicit input list
         }F               # inclusive range
    jd256                 # convert to list of base256 digits
 j\.                      # join by "." 
m                         # map over inclusive range 

PHP, 71 Bytes

Output as string

for($s=ip2long($argv[1]);$s<=ip2long($argv[2]);)echo long2ip($s++)," ";

Try it online! or Output as array

print_r(array_map(long2ip,range(ip2long($argv[1]),ip2long($argv[2]))));

Try it online!

Python 2, 128 bytes

a,b=[reduce(lambda a,b:a<<8|int(b),x.split('.'),0)for x in input()]
while a<=b:print'.'.join(`a>>i&255`for i in[24,16,8,0]);a+=1

Try it online!