I think this depends more on the implementation of the hashing algorithm than their ability to hash a null-terminated string including the null-terminator itself.
A few of such occurances of hash function implementations were observed in TrouSerS (version 0.2 built on Trousers Software Stack 1.1, or earlier) used in Gentoo Linux as one of the crypto libraries in its software stack and in TrustedJava (before jTSS 0.4a), and in IBM Client Security Software, both implementations included null-terminators on password popup dialog entries (null-terminated strings including null-terminators themselves).
Similarly, you can UTF-8UTF-16 encode cleartextplaintext before passing it to the hashing function, causing every other character in the input to be procesed as
0x00 or null character for non-extended characters by the function that processes input as plain ASCII. Depending on the hashing algorithm itself, its intended purpose, and the input length, this could be either beneficial (say, to password hash security, e.g. hashcat has some problems with these), or cause decreased password entropy, if that causes the hashing algorithm to ignore parts of the input (e.g. bcrypt, a cryptographic key stretching hash algorithm implementation based on Blowfish symmetric block cipher, limits cleartextplaintext to 72, 55, or 51 characters - depending on who you ask - and ignores the rest).
As for a general advice whether to include null-terminators in inputs to your hashing algorithms, I have none and you should be a bit more specific about the intended use you're hashing data for (are we talking of cryptographic hashing for password storage or HMAC here?), and will vary greatly depending on your specific needs and chosen hashing algorithm.