Does such a thing as question tags exist in French?

Yes, the closest equivalent is "n’est-ce-pas ?" which is much simpler as it stays invariable unlike the English form. However, it is not that much used nowadays and is becoming too formal and quite outdated, at least in France.

— Tu n’a pas mangé, n’est-ce-pas ?
— Si, j’ai mangé.
or
— Non, je n’ai pas mangé.

The casual and much more common way to prompt for a reply is to append si or non like this:

— Tu n’a pas mangé, si ?
— Si, j’ai mangé.
or
— Non, je n’ai pas mangé.

— Tu y vas, non ?
— Non, je n’y vais pas. (or the more casual: — Non, j’y vais pas.)
or
— Oui, j’y vais.

There are also more insistant / casual question tags like:

— Tu y vas, ou quoi ? (same as "or what?", similar to but stronger than the unbalanced "You’ll go to there, will you?")

— Tu y vas, pas vrai ? ("You’ll go to there, right?")

— Tu y vas, hein ?

— Tu y vas, d’accord ?

Does such a thing as question tags exist in French?

Short answer: no.

If not, how does one request an affirmative or negative answer to a statement?

The whole concept of giving a statement and then requesting an affirmative or negative answer is not something common in French.

When asked “As-tu mangé ?” or “Tu as mangé ?“, French people with naturally start their answer with yes or no anyway: “Oui, j’ai pris un sandwich” or “Non, pas encore“.

Finally, if you want to stress the fact that the yes/no element of the answer is really important to you, then you may start your question with “Est-ce que…“. Your interlocutor will then have to start their answer with a yes or no.

Est-ce que tu comprends les risques que tu as pris ?
Oui, je suis désolé.

I disagree with what everyone else has said so far. In my experience it is at least as common in French as in English for someone to add such a “question tag”. In particular, it is common in French to just add “, non?“. That might not be how French people write seriously, but it is certainly how many of them speak informally everyday.

(In my experience it is rare for someone to add “, no?” in English — that even sounds foreign (e.g. French!). Instead of “You’ve eaten, no?“, someone is more likely to ask “You’ve eaten, haven’t you?“.)