The 'ascension' and 'standing' of the pilgrim (before YHWH) (v. 3) must be in inner accordance with the (antecedent) 'coming/entrance' of YHWH (v. 7c; 9c). [*]In my opinion, this interpretation of the significance of נשא seems forced and Weber's interpretation of the broader relation between the stanzas themselves appears untenable.
1. It is hard to imagine that נשא in v. 4b can adequately represent the ethical requirements of the pilgrim. It is only one for four stated requirements, inconspicuously hidden in the middle of a tricolon.
2. Seen from a different angle, the יִשָּׂא of v. 5a could (on the traditional reading of the syntax of these verses, cf. Waltke-O'Connor, §31.6.2) certainly refer to the future consequence of the actions and states represented in v. 4. It relates however, to the whole verse, and not just to 4b.
3. Can שְׂאוּ/וְֽהִנָּשְׂאוּ represent Yhwh's entrance to the sanctuary? It could do, perhaps, but even if it did, it does not parallel the use of נשא in vv. 4-5, as there they predicate the one entering the sanctuary, whereas here they predicate the sanctuary itself. This is particularly troublesome if the supposed function of the repetition of this root is to highlight the parallel conditions that both entrants must fulfil.
4. The verb seems to have such different meanings in each occurrence. In the first occurrence (v. 4) it is part of an idiom about either using God's name in vain (cf. Exod 20:7) or worshipping false idols; in the second (v. 5a) it refers to the reception of blessing, in the final usages (vv. 7 and 9) it refers to the lifting of the heads of gates.
In short, it would seem that any direct connections between the uses of this root in Ps 24 are non-existent. Even at the auditory level there is no connection: the different moods (subjunctive/indicative, imperative), stems (qal, Nifal), conjugations (3rd person singular; 2nd person plural), and voice (active, passive) all sound very different.
Secondly: the broader relations between the stanzas (i.e. the perceived parallel in qualities required of both pilgrim and the Lord, the former following on the heels of the latter). Apart from the (IMHO) untenable view that God enters the temple before the pilgrim, there seems to be little that connects the requirements for their entrance. On the contrary, the two are strongly contrasted with each other. Whereas the pilgrim must fulfil ritual and ethical requirements as found in the Torah, God mustn't do anything, he's the King. When pushed, he lists his military attributes, which are hardly ethical requirements found in Torah. In addition to this, these attributes are not required by anyone, they're just stated as self-evident signs of authority and power. In the end, it would seem that it is not the attributes that gets him in but his special title: the Lord of Hosts (v. 10b). Again, even the nature of the dialogue is very different: the former consists of serene (or is it yearning?) question and answer, the latter consists of self-confident demand and resistant response.
So, we are still left with the question: can we attribute significance to the repetition of נשא, one that accords with a basic semantic sense purposely placed in each instance?