Replace 27.5 x 3" MTB tires with 27.5 x 2.1" Recently I bought a new MTB, a Frog Xtream Vantage. 

After two days of riding I realized a 3" tire size is too heavy and slow to ride. So I thought to replace the 3" tires with 2.1" regular MTB tires. 

Does that work? Will there be any damage to the rim?  

Also, will it improve my speed?
It mostly depends on the width of your rims. If, for example, your rims are 50 mm wide (internal bead distance), then 2.1" tires (nominally 53 mm wide) would sit very awkwardly on them. Instability at cornering and overall tire stability are likely to be problems. There are answers on this website about best tire width/rim width combinations, look them up if you are interested.

A better move would be to find tires which are just a bit narrower (2.8" or 2.6") and have lighter casing, or just plain lighter. The tires that you have right now may be just excessively heavy because they are cheap.