zsh, 603 594 566 561 548 440 415 399 378 370 bytes
ec
ho \\n;ca t<<<$'\x20';exi t
d$c -e8BC6P
d0c -eKp
$'\172\163\150' $'\055\143' $'\146\157\162 v \151\156 \173\043\056\056\134\175\175\073\173 \146\147\162\145\160 \055\161 $\166 '$0$'\174\174\074\074\074$\166\073\175'
$'\145v\141\154' $':\073\072\046\046\145\170\151\164';#%&()*+,/9=>?@ADEFGHIJLMNOQRSTUVWXYZ[]^_`jklmsuwy
0# $#;for b in {$..z};{ fgrep -q $b $0||<<<$b;}
Depends on coreutils + dc.
Try it online!
That was... a journey.
This answer has three parts. The first 4 lines handle certain special cases to simplify the code that follows. The next 2 lines and the last line both accomplish essentially the same thing, but exactly one is run with any given character removal. They are written with mostly complementary character sets, so that removing any character breaks only one at most, allowing the other to continue to function.
Looking at the first part, we first handle
- newline removal with
ec\nho \\n
- space removal with
ca t<<<$'\x20' (followed by exi t to avoid running later code, which would result in extraneous output)
$ removal with d$c -e8BC6P (8BC6 = 9226 is 36*256 + 10, and 36 and 10 are the byte values of the $ and newline characters respectively; we use hex digits in decimal to avoid having to include them in the large comment in line 6)0 removal with d0c -eKp (K gets the decimal precision, which is 0 by default)
In the next part, the only characters used (aside from the garbage at the end of the second line) are $'\01234567v;, space, and newline. Of these, four have been accounted for, so the remainder ('\1234567v) cannot occur in the last line. Expanding the octal escapes ($'\123' represents the ASCII character with value 1238), we get:
zsh -c 'for v in {#..\}};{ fgrep -q $v '$0'||<<<$v;}'
eval ':;:&&exit'
The first line loops through all characters used in the program and searches for each one in its own source code ($0 is the filename of the script being run), printing any character that is not found.
The second line looks a little strange, and appears to do the same thing as exit with a bunch of nops. However, encoding exit as octal directly results in $'\145\170\151\164', which does not contain 2 or 3. We actually need to make this less resilient to removals. This is because if any of '\014567v are removed, breaking the first line, the second line also breaks, allowing the remainder of the code to execute. However, we need it to also break if 2 or 3 are removed so that lines 3 and 4 can run. This is accomplished by shoehorning in : and ;, which have a 2 and 3 in their octal representation respectively.
The junk at the end of line 2 is simply there to ensure every printable ASCII character appears at least once, as the way the checking is done by looping through each one requires this.
If exit was not called in the first section (i.e. it was mangled by the removal of one of '\01234567v), we move on to the second, in which we must accomplish the same thing without using any of these characters. The last line is similar to the decoded first line, except that we can contract the range of the loop to save a few bytes, because we already know that all characters except for '\01234567v have been covered. It also has 0# $# before it, which comments it out and prevents it from producing extraneous output if 0 or $ were removed.
