Learn English – Can all transitive verbs take to-infinitive clauses

Quirk et al is a good grammar but weak, I think, on complex sentences.
What we're looking at in all of these examples is the remains of deceased clauses.

Of the four sentences, two:

• I saw her leave the room
• I heard someone shouting

are examples of special constructions that are limited to sense verbs, one with an infinitive and the other with a gerund. Note that you can swap infinitive and gerund here, with no particular difference in meaning:

• I saw her leaving the room
• I heard someone shout

All of these sentences have two meaningful verbs: see + leave and hear + shout. That means there are two clauses; every non-auxiliary verb is a clause. In this case the clauses are, logically:

• SEE (I, LEAVE (she, room))
• HEAR (I, SHOUT (someone))

In other words, the second clause is the direct object of the main clause that contains the sense verb -- they are Object Complement clauses.

Sense verbs are unusual, and have a number of syntactic peculiarities with their complement clauses:

• they can take all four kinds of complement clause
  (that-clause, embedded wh-question, gerund, infinitive)
• their infinitive complements mostly lack the to infinitive complementizer
• subjects of untensed (gerund, infinitive) complement clauses may or may not be present
• untensed complement subjects of sense verbs may undergo B-Equi, or B-Raising, or neither

So, the middle two sentences are simple complement clauses, enhanced by being complements of sense verbs, which have quasi-superpowers (only Operators like Modals, Quantifiers, and Negatives have actual superpowers)

Of the other two sentences, one:

• We knew him to be a spy

is, once again, an object complement clause (be a spy is a predicate noun -- SPY (he))

• KNOW (we, SPY (he))

Know can take a that-complement (with optional that, as usual):

• We knew (that) he was a spy

or an infinitive complement with B-Raising, and a normal to complementizer this time:

• We knew him to be a spy

There's little perceptible meaning in this distinction; only different syntax. The Raised subject in an infinitive clause can be passivized, for instance, but the subject in a that-clause can't:

• He was known to be a spy
• *He was known (that) was a spy.

And the final sentence:

• I got the watch repaired

is an idiomatic construction with get
(one of many such constructions; get has a number of meanings and uses in Modern English).

This is a causative get + Past Participle construction. There are two clauses, but their relationship is not obvious. The logical structure is something like:

• CAUSE (I, COME ABOUT (repair (indef, watch)))

or, in English:

• I brought it about (i.e, I caused it to come about) that Indef repaired the watch

That's rather a mouthful, and that's where get constructions come in.
One of get's meanings is the inchoative 'come to be' (get tired, get sick, get lost, get fixed),
and, like most English inchoatives, it can also be used as a causative 'cause X to come to be'
(get him tired, get yourself sick, get her lost, get it fixed). And that's what this one is.

It was a little bit of a challenge trying to deduce what you are asking for. I found a couple sites, which seemed to be of a bit of help. You're probably not going to find an all-inclusive list. I did find a lot of sites by simply searching on Google for "Verbs that must be followed by an infinitive."

Verbs that can be followed by an object and infinitive
Another site.