I bought the hyrule soldier armor in Hateno village (full metal heavy armor) and have been running around with it for at least 8-10 hours of game time. I have tried to carefully track its' side effects, since I also expected to move more slowly.
I just did some user tests to be sure, and this is what I can say for sure:
Running speed and stamina are not affected. Climbing speed and stamina are not affected. Dodging and jumping distance is not affected. These all work exactly the same. The speed is the same, and it consumes stamina at the same rate. I tried it with cloth armor and the hyrule soldier metal armor multiple times, there were no differences.
Even the noise meter was the same for sneaking. I had no issues with sneaking up on horses or enemies.
Metal armor doesn't even affect lightning strikes either.