The Doctor's Rummage Fail in "Smith and Holmes": he needs paper and checks through his coat pockets for it, digging up all kinds of interesting things in the process. Including Sherlock Holmes for Dummies.
Upon being properly introduced to Sherlock Holmes, the TARDIS immediately begins flirting with him. The Doctor's reactions are hilarious.
"I swear, Holmes, one of these days you're going to break your neck on these stairs! That booth isn't about to grow wings and fly off, for heaven's sake!" (The booth to which Watson refers is, of course, the TARDIS.)
Tesla's introduction to the series as the archetypical Mad Scientist is truly funny... until, suddenly, it's not. Still pretty funny while it lasts.
Holmes and Watson attempting to be normal, late-21st century human beings in "Child of Time". And early scenes with John Smith, in which he is nothing more than an Adorkable human version of the Doctor sans all the grief but holding on to the quirkiness.
Ten interacting with Nine. The Doctor just can't seem ever to get along with his other selves... Especially gold is Nine's commentary on Ten:
"What was I trying to do, impress a girl? ... I was? Aren't we a little old for that?"
Sherlock Holmes and Jeremy Brett getting drunk together in a pub. Yeah, it really does go over about as well as you'd expect. Very briefly, it's funny... and then straddles the line between Not Funny and Black Comedy.