Occasional Runner
That may be true, but it was not the question that was asked. The question was "Will just running on trails make you slower."
And the answer is yes. Running trails will definitely build strength, but that strength is useless without training your legs for maximum turnover, which you will not get on the trails.
Slower than what? Your optimal fast pace, probably.
I agree with Lace it is mostly about turnover. However, if I am running sub 6-6:30 minute pace, I do not care whether it is downhill or flat on a trail, or on the road. I could be faster with some track workouts, but I do not care for them and run about 98% trail. I do try to mix in some shorter workouts with a couple flattish, non-technical miles so I can turn the legs fast, do 1-2 climbs of 0.5 mile or so, and one technical descent where I let it hang out, test the turnover, jump obstacles, etc. A good hard hour works ok for me with that mix.
I also had a good run last week with about 5.5 miles and 500 feet vertical on non-technical trails where I averaged just under 7:00 minute pace overall and did a few turnover intervals. I need to worry about more long runs before focusing on more speed work with my goals, I think.