I will be travelling to Dubai from USA in a few days. I am an Indian citizen but I work in USA on the work visa. I recently got my UAE visa done online using an Indian travel agency. The place of issue on my visa says "Mumbai" since that is where the travel agency is based, but I have not physically been to Mumbai to get the visa done (its just an online visa).
Should I be concerned about any problems at Dubai immigration because of the place of issue on my visa document?
There is no concern about place of issue for your visa. UAE will be completely uninterested in that. (How would they know you've never been to Mumbai, anyway?).
While you never went to Mumbai, the agent who issued your visa never left Mumbai (at least not for issuing your visa). In times of telecommunications, it is perfectly possible for the issuing to take place at a different place than where you are, in particular if there is no such thing as a mandatory "face check" in the process. And it can be assumed that the UAE are aware of this technology.
This would be no different from applying for a visa from a country where there is no consulate or embassy. The application is then handled through a consulate or embassy in another country by mail, and they're the one issuing the document.
They want the place of issue mentioned so it's easier for them to contact that specific office in case they need more information during the customs process, which means less inconvenience for everyone involved (including you, as it may affect your time waiting), not to make your life harder.
The place of issue is about the passport/visa, not about you.
It is normal to have a different place of issue. But the place of issue is important to verify the visa/passport: it tells which authority has delivered it, so it is easier to check in case of problem (or unreadable parts). This was important on past, where someone should check in archives the documents (one do not want to call all embassies and authorities to know about visa [or getting from central authority based on the number, the issuer]). Now that all is centralized in a database, there is less need of such data.
Note: it helps also for plausibility: if the issuer of visa is far away (maybe a now closed embassy) or from a place where falsified document are common, an extra check is done.
