Is there a suffix like "phile" or "phobe" for don't care?

I am not aware of a suffix per se with that meaning, but I sometimes see the idea expressed by forming a pairing with the word agnostic. Although the first meaning of this word is specifically about religious belief, it can also be used in a more general sense:

2. a person who is unwilling to commit to an opinion about something

(Merriam-Webster)

Though M-W does not document it, other sources indicate that the word can also be used as an adjective to describe a person who holds the (non-)beliefs of an agnostic on a particular subject.

Thus, you might say "Chrome and Internet Explorer both have strong partisans, but I am browser-agnostic."

As far as I know, no. We could make one up, here. "-phile" and "-phobe" are derived from the greek words "philia" (love) and "phobos" (fear), so we'd want to look for an greek word meaning "indifference", I think.

"adiaphoria" looks like an early contender, being what Google translate comes back with for "indifference". There is already a philosophical concept "adiaphora" (note no 'i' at the end) meaning "indifferent things", referring to (my summary) things which are neither moral nor immoral. https://en.wikipedia.org/wiki/Adiaphora

This isn't quite what we're after, and it's a bit long, anyway - it would be nice to be able to have a single-syllable suffix like "phile" or "phobe".

"neutral" gives "oudéteros", which again is a bit long.

Perhaps we could use "mesis" which means "middle" - so you could have the adjective suffix "-mesic" like "phobic", and the noun suffix "-meso" like "phobe".

Eg, an "audiomeso" is someone who neither hates nor loves music, and if you genuinely didn't give a sh*t about whether someone is gay, you could call yourself "homomesic".

EDIT: @Nathaniel, in the comments, pointed out that the noun form suffix should be "-mese", not "-meso", so "audiomese" or "homomese". This feels right to me and I thank him for it.

I propose '-meh'

• Arachnophobe

• Arachnophile

• Arachnomeh

In the right tone of voice, I think that could work.

Meh, Wikipedia

Meh is an interjection used as an expression of indifference or boredom. It may also mean "be it as it may". It is often regarded as a verbal shrug of the shoulders

It seems you are looking for a concise, easily understandable term for someone who is neither a -phile nor a -phobe. I would suggest the suffix -neutral. It may be more of an adjective than a noun, but it can be pressed into service as a noun by ellipsis. Thus:

Which of the following describes you best?

I am a technophile

I am a technophobe

I am techno-neutral

90% of the technophiles, 50% of the techno-neutrals* and 10% of the technophobes said they had heard of the product.

(* where techno-neutrals is an ellipsis for techno-neutral respondents to the survey.)