So, yes, the title basically says it all. I've been thinking (and experimenting) a lot about this. Also, this is closely related to this thread:
https://www.pianostreet.com/smf/index.php?topic=57926.0 (and my response in it, in which I'm guilty of blindly accepting the paradigm of no pedal and even stating it as a personal opinion).
This sonata is notorious for its revolutionary use of the damper pedal in the third movement. And Beethoven only wrote pedal markings when he wanted an unconventional use of it. Hence, it is reasonable to think that when no pedalling is indicated, the player should use his/her good taste to choose appropiate pedalling in a "conventional" manner (i.e. if a chord repeats itself 14 times in a row without portato or staccato marks you hold down the pedal). So it seems to me quite weird that the accepted paradigm is that the first theme (first movement) should be played without pedal. And this includes when the theme appears in semiquaver tremolos.
By the way, von Bülow's comment on this theme in his edition seems spot-on to me (see attachment).
I have heard/can think of 3 arguments against using the pedal here, but, as I'll show, none seems definitive to me.
1. The last note in the right hand motives (e.g. g-b-a-g) has to be staccato. Of course this is true. Nevertheless, it can be done while using the pedal. Let me explain (referring to the first 3 bars): At first, I'd change pedal as usual (when changing harmonies). But when the staccato g occurs, I'd change again. If the g is very staccato and the pedal change is slightly delayed (as is usually done), the left hand while keep going as if no pedal change had occured, but the g will sound staccato.
2. In bars 261-2 and 265-6, the theme (in the left hand) should sound similar as it has throughout the movement. And I agree (it'd be inconsistent if all the previous times it sounded as a smooth, vibrating melody (as von Bülow puts it), and only here it was portato). This nearly beat me. I was nearly convinced that, because of these 4 bars, the first theme shoud always be played senza pedale. Nevertheless, today I tried this measures changing pedal every quarter beat (which turned out to be a tough coordination exercise because of the syncopation in the right hand octaves) and it works! Nevertheless, those octaves aren't marked legato (and they sond legato if played as I wrote, of course), so maybe this could be the definitive argument. But, to me, this does not sound as a crass mistake.
3. The first theme is as if string players were playing it with short bows (pa-pa-pa-pa...). If I didn't explain myself well enough, search "Jerome Rose Waldstein lesson" on YouTube. Well, this is a nice idea (as are many more analogies between passages in Beethoven's piano works and orchestral sound). But, correct me if I'm wrong, it is quite subjective. Perhaps this passage is (as are many more in the sonata) entirely pianistic? Perhaps the basic idea is the melody as notated in the attachment by von Bülow, which would not sound good in the piano because of the decay of the notes (but would sound well in a string section), so Beethoven found this "vibrating" solution (in fact, two vibrating solutions!)?
(4. It is harder without. If you suffered about making the tremolos sound even without pedal (see the thread linked above), it won't be easy to accept it if a student came to you and played the passage with pedal, which sounds so nice and even and is so easy! Well, point 3 here does pose a challenge. Nevertheless, this is of course not an acceptable argument (more like a joke), but could be the real reason behind the paradigm.)
So what do you think? I know we won't "solve" the issue, of course. It just seems weird that pretty much every recording of this piece (at least all I've heard) have no pedal here. In fact, the most radical approach I know of is Arrau's, who in his edition suggested changing pedal every quaver

(while allowing no overlap of sound).
By the way, I haven't stated something obvious: I like how it sounds with pedal (specially in the tremolos and including bars 261-2 and 265-6). Nevertheless, I also like it without.