Why would an op-amp use BJTs over MOSFETs?
The 741 is an old piece of junk, primarily used to teach basic electronics for cheap. I seem to remember reading somewhere that if every 741 ever made were to be collected, there would be enough to give every person on earth 6 or 8 of them. 

Modern op amps fall into several categories.


General Purpose - These op amps are not very fast, have bad non-ideal characteristics (bias currents in the nanoamps), drift, have input impedances in the megaohms, and cost almost nothing. The 741 falls in this category.
FET Inputs - These are a bit faster, have significantly better non-ideal characteristics (bias currents in the picoamps), drift very little, have extremely high input impedances (gigaohms), but may cost a few dollars.
CMOS - CMOS op amps are slow, but have excellent non-ideal characteristics (bias currents in the FEMTOamps), extremely high input impedance (TERAohms), drift about as much as general purpose op amps, and may cost a few dollars. This is the type of op amp that can get its output within millivolts of the rails, but rail voltage is limited.
Chopper Stabilized - This is another form of the CMOS op amp. It drifts very little, and has very low offsets. Take a look at this article for more information


There are other op amps out there that can handle RF frequencys, or handle high output currents, but they don't really fall in these catergories.

As you can see, each type of op amp has different non-ideal DC characteristics, and input impedance. How much current flows into the op amp inputs depends on the input impedance. For most modern op amps, these are very small currents, and can be considered negligible for the majority of applications. Which type of op amp you use is a design consideration, factoring in speed, cost, temperature range, and any precision concerns.