Seem to do;seem to be doing

Well, of the four choices, B. is automatically out. to-infinitive verbs do not carry tense: "to felt" is made of up a past tense verb 'felt'. The correct form for B. would be "to have felt", which is C. And that's out too, because if you look at the time sequence, "last week", it's specific and the present perfect "have felt" is not compatible with adverbs expessing specific time. So, both B. and C. are out as your question implies. Now, A. is also out because 'seems', a linking verb, links a to-infintive verb, "to feel" with "tired":

A. He seemed to feel tired.

to-infinitive verbs express an event as having not taken place yet, whereas -ing verbs express an event as having already taken place, so if we add -ing to the bare infinitive, we get "to be feeling", which is D.

D. He seemed to be feeling tired.

'to be feeling tired' means, the event took place.
'to feel tired' means, the event hasn't yet taken place.

All the best,

Actually, "last week" is compatible with "seemed", which is not in the present perfect tense, so there must be a different reason why C is wrong.

"He seemed to have been confused last night", for example, is perfectly correct. Isn't it?

I wonder if the verb "feel", a 'private verb', is involved in the explanation of why C is wrong.

I would (tentatively) parse the problem slightly differently:

1. She read the letter again. He seemed from the words on the page to have felt some grief when his mother died; but who could be sure.

OK. So can we substitute a noun of physical feeling for ‘grief’?

2. She read the letter again. He seemed from the doctor’s account to have felt merely very tired, at first; only later did the chest-pains begin.

Seemingly. Can we now introduce a specific adverbial time-phrase comparable to ‘last week’?

3. He seemed from the doctor’s account to have felt merely very tired, that Tuesday; only in the morning did the chest-pains begin.

(I would say, by the way, that although ‘last week’ wouldn’t be ok with a present perfect verb, it would be ok with a perfect infinitive:

4. He is supposed to have visited his father last week.
5. He ought to have gone last week, but he didn’t.

I think too that ‘seem’ is ok with a non-linking verb:

6. He seemed to like writing extremely long posts on English Forums.)

So if ‘seemed’ + ‘perfect infinitive’ + adverb of time + physical feeling is permissible, there must be something else wrong with C.

Looking again at A and C:

A:
7. He expected to feel very tired last week [but didn’t].
– fine.

8. He seemed to feel very tired last week.
– not comfortable; ‘last week’ implies a lengthy, continuous state of feeling; we want to make the verb continuous (= D). Yet:

9. He seemed to feel very tired at about 6o/c, doctor; then it seemed to pass.
– fine.

So perhaps with A, the problem isn’t grammatical, but semantic.

C:
10. He seemed to have felt very tired last week =
11. Last week, he seemed to have felt very tired.

i.e. last week, we inferred that ‘he’ had, at some point previously, felt ‘very tired’! Does the sentence strike us as not permissible simply because its meaning is so remote? Cf:

12. Last week, he seems to have felt very tired.

This is fine: we’re talking about our past perception of his feeling tired; not our past perception of his having previously felt tired.

So perhaps the problem in both A and C is one of meaning, rather than grammar.

MrP
PS: All of which is the long way round to CJ's 'private verb', via Casi's progressive.