Jesus could have in mind a physical place in the Synagogue where there would be read out "the Pentateuch and the Prophets."1 The Jewish Encyclopedia has an entry for this under Alemar which is a corruption of Al-Minbar (Arabic for 'the chair [or pulpit]').1
The above-mentioned Jewish Encyclopedia (JE) entry states:
"In Russia it still goes by its Talmudic name בימה [i.e. בִּימָה bimah],* which is simply the Greek βῆμα [bḗma], a speaker's tribune"
[*] Transliteration as well as emphasis mine.
The entry in the Liddel & Scott Greek Lexicon for βῆμα [bḗma] is:2
βῆμα, Aeol. and Dor. βᾶμα , ατος, τό, (βαίνω)
A. step, pace, h.Merc.222, 345, Pi.P.3.43, A.Ch.799 (lyr.); “σπουδῇ . . βημάτων πορεύεται”
E.Andr. 880; “τοσόνδε β. διαβεβηκώς” Ar.Eq.73; footfall, “ἐρατὸν βᾶμα” Sapph. Supp.5.17; Διὸς εὔφρονι βήματι μολεῖν to journey under the kindly guidance of Zeus, S.El.163 (lyr.); gait, “β. οὐκ ὀρθόν” Hippiatr.27.
step, as a measure of length, = 10 παλαισταί, about 2 1/2 feet, Hero *Deff. 131.
metaph., step, 'moment', “πρόοδος ἐν τρισὶ β. διισταμένη” Dam.Pr.258.
II. = βάθρον, step, seat, S.OC193 (lyr.).
- raised place or tribune to speak from in a public assembly, etc., Th.2.34, LXX Ne.8.4, etc.; in the Pnyx at Athens, ἐπὶ τὸ β. ἀναβῆναι enter public life, D.18.66; “αἱ ἀπὸ τοῦ β. ἐλπίδες” Id.4.45; also in the lawcourts, Id.48.31, Aeschin.3.207; of a suppliant, “ἐπὶ τοῦ β. καθεδούμενον” Ar.Pl.382; in the βουλευτήριον, Antipho6.40.
b. tribunal of a magistrate, “τοῦ ἡγεμόνος β.” PTeb.434 (ii A. D.).
- = θυμέλη, Poll.4.123; “β. θεήτρου” IG3.239.
- base, pedestal, OGI219.36 (Ilium, iii B. C.), 299.15 (Pergam., ii B. C.).
But of course Christians might be more familiar with the so-called 'Bema-seat' of Christ, that is, His Judgement Seat:
2 Corinthians 5:10
For we must all appear before the [Judgement] Seat [Gk. τοῦ βήματος] of Christ, that each of us might recieve according to what he did in the body—whether good or bad.
Inasmuch as Jesus singles out "the scribes and Pharisees," (Mt 23:2) He singles out these as those who have the authority of 'binding and loosing.' He probably leaves out the Sadducees as they were heretics who denied the resurrection, only held the Pentateuch to be Scripture etc. The JE says that "they broke away from the Law and lived in great luxury, using many silver and gold vessels at their banquets; and they established schools which declared the enjoyment of this life to be the goal of man, at the same time pitying the Pharisees for their bitter privation in this world with no hope of another world to compensate them."3 As for Rabbis, the orthodox ones would have been Pharisaical.
The JE states that, quote:
The power of binding and loosing was always claimed by the Pharisees.4
The same entry says in the section In the New Testament, that:
"Jesus ... found [this same power] belonging to the scribes and Pharisees".4
However, I think it refers to the 'office' of the legimate, God-ordained teaching authority of the time, and not a physical chair. For one thing, there is no record of a chair of Moses being preserved or lost, leaving it to be an office which at most might involve a physical chair.
While Matthew doesn't use the word βήματος (but καθέδρα) this is not important. After all (as you can see from the lexicon entry above: II:2), ἐπὶ τοῦ βήματος καθεδούμενον (being seated upon the βήμα/tribunal seat), you can and do sit in or at a 'βήμα.'
You may have heard of the word 'Cathedral' which names signifies a church wherein the bishop's 'chair' or καθέδρα (throne/chair) is located, where he presides over the solmen functions of the church.
This undoubtedly has its origin in the 'chair of Moses,' to which this place in a Synagogue seems to have been the referent. In that in the New Dispensation, the New Covenant, the bishops take the role of the teachers of the Church, over whom, since the New Covenant, Jews do not have authority whatsoever.
The 'seat-position,' in any case, is a position of authority. Jesus calls it the seat of Moses, i.e. Moses, the authoritative teacher and interpreter of the Law of God, able to bind and loose.4
[More specifically, in reference to the Bishop of Rome (a.k.a the Pope) or an equivalent authority (an Ecumenical Council backed by his authority) when he judges ex cathedra, that is, 'from the chair,' it signifies an infallible judgement or teaching. Of course, he doesn't have to be literally sitting in any chair to pronounce such a judgement; it refers to an authoritative office.]
Tertullian, a Western Father, speaks of the episcopal succession of the Apostles, the different Sees (an area subject to the jurisdiction of a certain bishop) of the Church, as cathedrae apostolorum (the [episcopal] chairs of the Apostles):
Age iam, qui uoles curiositatem melius exercere
in negotio salutis tuae, percurre ecclesias apostolicas
apud quas ipsae adhuc cathedrae apostolorum suis locis
praesident, apud quas ipsae authenticae litterae eorum
recitantur sonantes uocem et repraesentantes faciem
Come now, thou who wiliest to exercise thy
curiosity to better purpose in the business of thy
salvation: go through the Apostolic Churches
where the very thrones of the Apostles at this very
day preside over their own districts, where their
own genuine letters are read which speak their
words and bring the presence of each before our
1 1906 Jewish Encyclopedia, A, Alemar.
2 Henry George Liddell, Robert Scott, A Greek-English Lexicon, entry for βῆμα.
3 1906 Jewish Encyclopedia, A, Sadducees | cf. Mt 22:23; Acts 23:6-8
4 1906 Jewish Encyclopedia, B, Binding and Loosing | cf. Mt 16:18-19; 18:17-18; Lk 10:16; 2 Cor 2:10b
5 De Praescriptione Haereticorum, XXXVI.