What will happen if I export a 720p video in Photos to 1080p? I'm considering exporting all the videos currently stored in my Apple Photos library. These are made up of a mix of different video "qualities," including 1080p and 720p.

When I choose the File > Export > Export [number] Videos command, a dialog box appears prompting me to specify a quality of the exported files, e.g., 720p, 1080p, etc.

If I choose 1080p, I presume that my 1080p videos will be preserved in a lossless way. That is what I want. 

However, I don't understand what will happen to my 720p videos. Will the software export them in a lossless way as 720p, i.e., keep the files as-is, or will the software do something else? Will it try to re-encode them to "up" the 1080p format, potentially increasing the file size?
You've only got so many 'real' pixels to play with, so when you stretch that out to 1080p, you've got quite a few holes.

From there it all comes down to what approach the software you are using takes to create the missing pixels.  Typically the stretch it out to the edges, and then interpolate back to fill the holes, based on what the two pixels look like either side of the gap.

Some software does this better than others, and the best method to use can depend on the image content also.  Typically it makes things look softer, blocker if bad, and highlights poor quality issues such as moire etc.

Premiere Pro includes a simple algorithm as standard, and you can use plugging more targeted to the task if you want also.
There's no right or wrong.  A full HD TV will be doing this 720p to 1080p conversion ever time you watch anything less than full HD anyway for example.

Here is a simple graphics showing what happens