Even if you have a valid residence permit from UAE, you can still acquire a residence visa in Qatar. Only that, your HR will have to probe that you have no intentions of residing on ANY of which BUT ONLY in Qatar. In short, as they stamp the Qatar visa-it means that you choose to reside in Qatar not in UAE.

The embassies will not deal to cancellation matters because it is the immigration who deals with expatriates.

Nonetheless, dont worry about it. Its not much big of deal. If you dont know someone from your company who can handle your passport with full dilegence and promise to return it on time. You'd rather ditch the UAE visa and pursue whatever Qatar visa you.