Will leaving 1 of the 3 GND pins on the ATMEGA328p ungrounded cause any problems?
Depends on what that pin is for. It may be required to link two grounds in the silicon, in which case it would cause an issue. You say it is linked, so it shouldn't be that bad. But ground pins are used for two other reasons.

Thermal: a ground pin gives a nice thermal link to a large copper plane which can be used to suck heat out of the IC and stop it over heating. Not connecting it up will cause there to be more heat in the IC, possibly spot heating, shortening the life of the IC as well as possibly causing damage straight away.

EMC: ground pins give you more routes for the current loops to follow. This reduces the chance of EMC problems and means that the IC should perform better. The ground link also reduces the inductive path for the current, allowing faster signalling and general improvement in performance.

As mentioned in the comments, and which I completely forgot about, the extra ground pins may also be carrying extra current. For every bit of current in, there must be a bit of current out. It can be surprising just how much current needs to flow in/out of these ICs.

So, while not connecting the pin may not cause immediate issues, it may cause a lot of problems down the line. If your design does not let you connect the pin, I would suggest your design needs to change so that it can. Connecting a ground pin is an easy task compared to working out why the whole product fails at some random point.