Both are fine, and I agree with both Jonathan and Anelisiwe. But I wouldn't use either of these in normal speech. I would say one of these things: I don't eat sugar anymore. I'm not having/eating sugar anymore. I took sugar out of my diet. I'm off sugar. I don't eat/have anything with sugar in it anymore. (Brits can drop "it" here). I removed sugar from my diet.

You can only "eliminate" sugar from your diet if it was part of your diet. You can "exclude" sugar whether or not it is actually part of your diet.

Eliminate means completely removed, meaning that you have completely removed sugar from your diet, it is no longer there. Excluded means not including something in a particular category, meaning that your diet does not include sugar at all.