What is the widest tire that I can safely fit on a rim that measures 700x19.6 (outside)x14 (inside). It's got 25's on it now. I'd like to go up to 28.
The (inner) rim width is measured this way:
and for a 13 mm width, 18-25 mm tires and for 15 mm width, 23-32 mm tires. So, So, a 14 mm width (assuming you're measuring properly) should be fine with 28 mm tires; you can exceed the recomendations by a bit usually without adverse effects.
I habitually run much wider tyres than that on narrow rims. My shopping bike has a 559 rear wheel with an Aeroheat rim that's 24mm wide. I run a "Fat Frank" 55mm tyre on it. That is, however, a low pressure tyre (4bar/60psi).
On 406 wheels it's much more dramatic. I run high pressure flatland tyres 55mm or so wide at 6bar or more (100psi+) on my load bikes, and the rims range from 28mm on the back of my touring bike to some generic Taiwanese aero rims that are about 22mm wide.
The answer is that once the tyre is Ʊ shaped at all it doesn't seem to matter. If you're running a 25mm tyre on a 19mm rim, you're already putting all the outward force on the rim that you're going to put, regardless of tyre width.