How long can I leave lights on a planted tank on?

8-10 hours is suggested for plants.

plants will easily tolerate 12+ hours of light per day.

The only reason for most to reduce lights on time is to reduce Algae growth, so if you need 12 hours use 12 hours. what is missing from this all however is light intensity, just how much light are you planning on or working with, and over what area, and to what depth, it all factors in.
But as a general rule if it is not heating your tank water up, 12 - 14 hurs is not out of line. Not needed for the plants, but it wont hurt them either unless your in the 3+ watt range, then you might wish to make some changes/additions.
Like adding CO2, increasing Ferts etc etc etc

There's no reason to have the lighting on for 12 hours a day.

I don't think you can't directly compare photoperiod with light intensity though. 12 hours with 30 watts is not the same as 6 hours with 60.

7-10 hours for plants, I have a photo period of 8 hours in my tanks.

They can do 8-12 hours. Although I like to do only 8 hours a day, sometimes 10. It really depends on my mood. Going over 10 seems a little excessive to me and it generally starts algae problems if it's a medium-high light tank.

Low light tanks are able to tolerate longer photoperiods but even then, eh... it's iffy. I don't see the point in keeping it on so long anyway. Waste of electricity

What kind of diatoms? Green spot algae? If that is the case, higher lighting will help a lot. How long the lights are on doesn't seem to affect growth so much and would end up encouraging other algae to grow instead. Green spot is pretty normal to have, to a certain extent.

Oh, brown? Ugh, that stuff is annoying.

Easy way to trigger a brown diatom bloom would be to just stir your tank up a lot and cause a mess. If this is an established tank, which I would assume it is already, then increasing light intensity or hours won't do much. Causing a mini-cycle can trigger brown diatoms to appear too.

Not sure why you'd want to do that though. It'll cause more problems than it does help your otto's diet. Algae wafers and other natural algae in the tank should suffice in terms of keeping them happy and healthy.