Do I need to supply an op amp with more voltage than I need it to output?
Maybe

Some opamps can provide an output which is very close to or even at the supply input voltage (known colloquially as the "rail"). These are often known as rail-to-rail opamps, and are typically slightly more expensive than standard ones.  Normal ones need some extra voltage, anywhere between a few hundred millivolts on a low-voltage opamp to 25V on an opamp designed for 1kV use.

There may also be some opamps out there which have a minimum supply voltage other than that described above.  If so, I haven't seen one. But it'll say in the datasheet.

The only way to answer this question is to check the datasheet for the opamp you plan to use. You can edit your question with the part number or (better) a link to the datasheet if you're not sure how to interpret it. Or, if you haven't picked an opamp yet, buy a rail-to-rail one.