Based on your description of what you were given, then you should not need to contact DMV/DOT
However if you are not comfortable with that answer then contact/visit the dealer, you can ask them how long it normally takes, and do you pick up the real plates at the dealer, DMV, or will they be mailed to you. Also call them if the temporary plates will run out in a few days, to make sure everything is good to go.
One other note. If the dealer is in one state, and you live in another they can give you temporary plate for their state but may not be able to file for the real plates in your state.
Once everything is finalized go online to DMV and make sure that the car registration is OK. A few years ago the dealer gave me real plates, they gave me a registration good for two years. But the info sent to DMV was corrupted: the VIN was in the system, but the description was wrong and the plates were listed as none. This was only noticed when I tried to re-register the car two years later. In fact according to DMV the plates on the car were listed as never issued. If I had ever been pulled over it would have taken hours to resolve.