If your lifters are tapping only when you engine is cold, the mechanic technically is right when he said "nothing's wrong" as far as Mitsubishi is concerned. For example, lifters (technically referred to as lash adjusters by Mitsu) had a very poor design in their '90-'99 Eclipses. You would hear tapping whether the engine was running for a minute or an hour. It really didn't hurt the engine at all, which is why Mitsu never recalled them, but the excess tapping would sometimes trigger the knock sensor to go off, hence the car's computer would retard the timing so maximum horsepower could not be reached.

It is hard to avoid the lifter tapping all the time, but you may want to try using a high-grade synthetic oil. During the summer, I would recommend a full-sythetic oil with a weight of 10W-30. Use 5W-30, if your winter months' temperatures fall below 32 F. Also, if you use a synthetic like Mobil 1, use the Mobil 1 filter. It is more expensive, but they do a great job.

I don't think money is an issue for the mechanic to change your lifters. It is actually a pretty easy, and fast job. Plus, the mechanic gets paid whether or not he replaces your lifters. Most dealer mechanics get paid a salary, not by an hourly commission.

If your lifters quiet down after a few seconds (or even a few minutes) of driving, you shouldn't worry. Your car is fine. If they never quiet down or seem to get noiser, you may have a problem. Take your car right to the dealership and leave the car running while the tapping is occuring so the mechanic can hear the noise for himself.
Good luck!