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According to a few tutorials I am seeing, DOS used dollar-terminated strings to write to the terminal. This seems to also be documented here on INT 21H.

AH = 09h - WRITE STRING TO STANDARD OUTPUT
Entry: DS:DX -> '$'-terminated string
Return: AL = 24h
Notes: ^C/^Break are checked

You can see this from the code in this tutorial here. What was the reason for going with dollar-terminated strings rather than NUL-terminated strings like C?

The short answer is that DOS was designed to be similar to CP/M, and drawing a quote from here:

While 8-bit programs could not run on 16-bit computers, Intel
documented how the original software developer could mechanically
translate an 8-bit program into a 16-bit program. Only the developer
of the program with possession of the source code could make this
translation. I designed DOS so the translated program would work the
same as it had with CP/M – translation compatibility. The key to
making this work was implementing the CP/M API.

Or course this brings up the question as to why CP/M used the dollar sign.

This discussion says CP/M got the idea from DEC, which used the RAD50 character encoding. With only 40 characters (50 octal), you only have uppercase, digits, space, period, dollar, and percent.

Both CP/M and RT-11 are evolved from earlier DEC OS's, most notably
OS/8 (on the PDP-8) and DOS-11 (on the PDP-11). The most obvious
feature of all of these OS's is the presence of "PIP"

So DEC probably chose dollar because it didn't have many options, CP/M got it from DEC, and DOS got it from CP/M.

So, what about the last part of the question, why not use nul as c?
As shown in an earlier answer, MS-DOS was influenced by CP/M, CP/M was influenced by DOS/11 on PDP-11 and that was an extension of OS/8 from PDP-8.

OS/8 used a character set called DEC Radix-50.
After checking the link about the RAD50 character set, the answer why OS/8 didn't use nul to terminate strings is quite simple. nul is Space in that character set.
Considering that apart from letters and digits you only had 3 punctuation marks ".", "%" and "$" you had to choose one.
The question remaining is why they choose $ and not %.

Besides that, as commented below, C did not exist when the PDP-8 was constructed, so it could not have influenced the choice of string terminator.

It's worth noting that CP/M was written originally in the PL/M-80 programming language (also developed by Gary Kildall) (source here )

The PL/M-80 Manual on page 11 states that

The character set used in PL/M is a subset of both ASCII and EBCDIC character sets. The valid PL/M characters consist of the alphanumerics

A B C D E F G H I J K L M N O P Q R S T U V W X Y Z

a b c d e f g h I j k l m n o p q r s t u v w x y z

0 1 2 3 4 5 6 7 8 9

along with the special characters

= . / ( ) + - ' * , < > : ;

and the blank characters

space tab carriage-return line-feed

If a PL/M program contains any character not in this set, the compiler may treat the character as an error

That's 80 characters.

Interestingly there is no mention there of $ at all - yet on page 15 it is listed as a character that "may be freely inserted between the characters of a constant to improve readability".

I can't say whether or why this might have affected the decision to use $ as the string terminator for function 15 in CP/M, especially since page 16 is explicit that "character strings are denoted by printable ascii characters enclosed by apostrophes" which doesn't limit it to the same character set as the code body.

I can say from personal experience that function 15 was almost never used, and CP/M / CP/M-86 / MP/M / Concurrent DOS / CDOS /… assembler programmers pretty well invariably used subroutines containing loops that called function 2, usually using either a byte count or a null terminator.