Yep, I've got a low-end mountain bike I use for most of my commuting - particularly in winter. Trek's bottom-of-the-line high-tensile steel mountain bike (the seat tube is chromoly - big deal), with Shimano's bottom-of-the-line Acera drivetrain. It's been 100% reliable in a few thousand miles of commuting.
1) First I got some 1.25" slick tires for $10 each from Performancebike.com or Nashbar.com. This is the biggest performance upgrade you can make. Weight is irrelevant on flat ground (physics says that once you get up to traveling speed it doesn't matter what the bike weighs). And it is not noticeable on uphills unless you drop several pounds off your bike. Not something that changing derailleurs will accomplish.
2) If you want further efficiency, you can get a rigid fork. My front shock weighed 5 lbs when the bike shop took it off! The Tange Chromoly Mountain Fork weighed just 2 lbs. However, weight savings was only part of the benefit. You stop wasting energy on bouncing the pogo stick - instead it goes into moving you forward. This is still minor compared to getting 1.25" slick tires. For $40-$50 though, including install, it's not an expensive upgrade.
3) Get fenders if you plan to be an all-weather commuter. The full fender variety can be a bit rattley, but keep you very dry when roads are wet. Easy-to-remove alternatives are (
http://mtbr.com/reviews/Extras/product_121896.shtml) and (
http://mtbr.com/reviews/Extras/product_121897.shtml). Those will keep your butt dry and mud out of your face, but won't keep nice clothes completely clean.
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4) Make sure you can be seen. Get some rear blinking lights (Planet Bike Superflash or Cateye LD1000 are the most recommended on this board). Some Scotchlite reflective tape can be had from Ebay or from your local autoparts store. I use the high-intensity silver/white variety. It is the brightest they make and very inconspicuous on metal bike parts in the day time. At night, however, it glows extremely brightly in car headlights (or camera flashes). About $5 for a three-foot roll at AutoZone or similar places.
Clipless pedals can be more efficient, too, if your ride is long enough and has stretches free of traffic lights and stop signs. I use them on my road bike (my "fair weather" commuter), but not on my mountain bike (my everyday commuter). The dual pedals are a nice way to get the best of both worlds, but are more annoying to clip into than straight clipless pedals.
If everything else on your bike functions, I'd leave it alone. Fancy parts won't make your commute any faster or more comfortable. But they will make your bike disappear a lot faster though when thieves see it!