I work in Company A and I'm kind of the star there (I feel it is mainly due to political reasons). Recently, I received an offer from Company B and I decided to accept it. HR department in my current company pushed me to review my decision and propose conditions under which I would like to stay at company. 

I feel that offer from Company B is much better for my career (project is more interesting, company is also a better suit for me), but I don't want to burn bridges. How can I respond?
['https://workplace.stackexchange.com/questions/78863', 'https://workplace.stackexchange.com', 'https://workplace.stackexchange.com/users/-1/']
> 
> I decided to accept it.
> 
> 
> 

If you've already made your decision and accepted the other offer, it's OK to say that.

> 
> I have already accepted the offer from Company B and thus cannot consider any counter offer. My last day will be XX/XX/XXXX.
> 
> 
> 

The expectation is that once a job offer is accepted (in writing), that that's it. You wouldn't accept an offer and keep interviewing or keep negotiating the terms so it shouldn't offend your current employer that you aren't going back on your word to Company B. It isn't burning a bridge to be honest here.
The best thing here is to put it back on the company. 

> 
> I am not comfortable trying to define what sort of offer I would be
>  comfortable requesting. I would entertain an offer by the company if 
>  it feels that I am under-compensated currently, or should be advanced to 
>  higher position, and consider it appropriately.
> 
> 
> 

This gets the onus of action off of you, and places it it back on the company. If they make you an offer great, you can consider it. But as you said you are looking at more than just compensation. At the same time there might be a number possible that the company could offer you that might make you reconsider. Either way it should solve your problem of not wanting to burn bridges.