Would using an application fee waiver affect my admission result in a negative way? I am applying to an Ivy League university in the US as an international student for a Ph.D. program. They have application fee waivers for students with financial hardship.

Would using it (or even asking for it) have negative effects on the result of application?

They also have a GRE waiver for these type of students too. I believe this one does affect the result. Am I right?
The people in the program will evaluate and score your application. 

They won't know whether the fee was waived, nor are they likely to care.

They will know if there is no GRE score. For some programs the GRE score is important, for others it is not. 

You will need to ask the programs that you are considering what there policy is. GRE waivers are usually in lieu of other evidence (such as a strong work history, previous graduate degree, performance on some other graduate test (MCAT, GMAT, etc)).