I have recently stumbled across the word どうしたって which according to the dictionary means "by all means, no matter what, at any rate". I was wondering how its usage differs from 是非 "certainly, without fail" as they both have very similar meanings. Does one have a positive connotation and the other negative? Are they interchangeable or not? If anyone could tell me the difference I would greatly appreciate it.
You are close. These are used in different types of sentences.
どうしたって (or どうしても) is usually followed by something undesirable or inevitable.
No matter what, it's impossible.
At any rate, it's hard to understand the feelings of other people.
是非 is an adverb that is used exclusively when you strongly invite or recommend something. I feel it's weaker than 必ず ("without fail"), though.
Please come to our party!
You should definitely play this game.
Yes, please do! / Definitely!
(after being asked whether they should do something)
(Note that both どうしたって and 是非 have other usages.)