I'm working on my commercial multiengine rating. And I got this question from my examiner, and I answered: landing gear produces penalty drag, so it decreases thrust. But the examiner says this isn't correct... ?
Thrust is not decreased. The drag penalty means that more thrust is required to achieve a performance condition than without the drag penalty. In level flight, more thrust is required to go the same speed. When climbing, more thrust is required to achieve the same climb rate. Your maximum thrust available is what the engine can produce, so the extra drag subtracts from whatever surplus thrust is available.
In a light twin the key thing is the surplus thrust margin available for climb at the most efficient speed for climbing while single engine (blue line speed). To fly in level flight single engine at blue line speed, we'll say you need 500 lbs of thrust from the good engine. To climb at that speed requires more than 500lbs, and to get a decent rate of climb, say 400 fpm, you might need 800 lbs (about what you'll get with a Lyc O-360 running flat out), which we'll say is what the engine makes at full power at sea level (a piston twin that can climb better than 300 fpm single engine at gross is a pretty good performer).
Add more drag by leaving the gear down, and the thrust required just for level flight goes up from 500 lbs to, say, 700 lbs because of the 200 lbs of drag of the gear sticking out. Instead of 300 lbs of surplus thrust margin available for climb, you only have 100 lbs and it's a struggle to get any climb at all or to avoid descending, and only if you use perfect technique (on-speed, 5 degrees bank into the live engine).
So the proper answer for leaving the gear down is, surplus thrust available for climb is decreased and therefore climb performance at maximum thrust is decreased. Which in a light twin at gross on a hot day means, as the old saying goes, "the remaining engine takes you directly to the scene of the crash".