I let him do it.
I allow him to do it.
Why does the latter require to?
What are the "rules" of using to with an infinitive? When is it necessary?
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In both OP's examples the verb do is an "infinitive". The basic rule is the infinitive is "marked" by to, but there are exceptions, notably...
1: Verbs of perception (feel, hear, notice, see, watch)
"I saw him do it"
"I let him do it"
"I made him do it"
If the primary verb isn't covered above① (and isn't an auxiliary/modal such as can, may, must, shall, will) the infinitive requires to...
"I permit you to do it"
"I ask you to do it"
"I encourage you to do it"
See here for more information on using the infinitive without to.
① I just copied a list of verbs from the site I linked to, but it seems incomplete; "I helped him do it" is at least "valid", even if some might prefer to do. Plus there's "I had him do it" (where no-one would endorse the "marked infinitive" form).