Most luthiers and repairmen will shy away from anything that's not lemon oil (Though other suggestions from luthiers include linseed and mineral oil). This is what people have been conditioning fretboards with for a long time and generally consider it safe when used sparingly and only on the wood.
Oiling a fretboard is considered purely cosmetic, however you'll meet more guitarists who prefer a "clean", "shiny" or "smooth" feel over something that has never been enhanced with moisture.
Here's a take on oiling the fretboard by Bob Taylor, creator of Taylor guitars. Most guitarists have this opinion on oiling a fretboard/unfinished wood parts.
Claudio, I’d have no worries about using lemon oil on my fretboard. It’s safe. Use it only on the unfinished wood like the fretboard and bridge. The wood can dry out over time, and an oil like this, or linseed oil, or even mineral oil, can protect the wood and beautify it as well. Don’t overdo it. Once a fretboard has been oiled a few times, you can slow down the frequency. The nice thing about lemon oil is that it cleans while it oils, so it won’t build up as easily, but be sparing. I don’t think your fretboard will need oiling more than twice a year, and eventually, once a year.
Collings, makers of beautiful and incredibly expensive guitars also suggest oiling an unfinished fretboard.
If you must use commercial products, avoid those with solvents, silicones or abrasives. Remember: polishing is not cleaning. Polishes remove finish along with dirt. Fingerboards can occasionally dry out, but require only a very small amount of boiled linseed oil (thoroughly buffed) to restore. Less is always best.
In general, everyone suggests oiling unfinished fretboards. This usually means Ebony and Rosewood, (however some enjoy doing it to maple to) more because they like the feel or look of it more than the guitar needs it.
With proper care, by the time your guitar makes it to you it's considered quite stable and the wood moisture content is in a happy place for all parts of the guitar. Your guitar shouldn't need anything on the fretboard if it's been well taken care of.
Also, as noted by Bob Taylor, one or two applications really is all a fretboard needs before it looks and maintains a great balance of moisture. Once you do it, you'll notice yourself needing to do it less and less.
P.S. Your guitar (most guitars) has a rosewood or ebony fretboard. This wood naturally has oils in it, even after curing, that help maintain it's stability and longevity.