It’s a bare infinitive (a.k.a. the simple form of the verb).
- What’s the subject of your main clause? I
- What’s the primary verb in the main clause? overheard
- Is the verb transitive or intransitive? transitive
- What’s the direct object? ?
Ay, there’s the rub!
The direct object is the phrase, a visitor remark,
“You know, that grandfather clock actually runs.”
Things get a little weird when a phrase with a subject and a verb
acts like a noun.
We have a list of how to handle those
at When should a verb be followed by a gerund instead of an infinitive?
That page doesn’t mention overhear,
but it does state that hear (and see) are
followed by either the simple form or the gerund (no “to”).
It may make more sense if you consider shorter sentences:
- I heard him sing.
- I heard him sing the anthem.
- I saw him play.
- I saw him throw the ball.
- I overheard him remark blah blah blah.
As indicated above, you can also use gerunds for these verbs:
- I heard him singing.
- I heard him singing the anthem.
- I saw him playing.
- I saw him throwing the ball.
- I overheard him remarking blah blah blah.