Do I really have to help pay for my friend's birthday gift?
Why wouldn't you chip in for their gift?
Because I just finished a major project and don't have any money left.
If you are unable to pay for your share of a group gift, you need to communicate that with people. If you don't tell them, they will just think you are selfish and rude.
That's embarrassing though. Instead I'll just pretend like I paid my share.
I understand that you would be embarrassed, but if you lie about it, eventually they will find out you didn't pay. Then they will confront you about it and things will be even more awkward. It's best to just be upfront and let them know you are struggling financially and can't afford to contribute.