were there any social classes (equestrians, senators) during the Roman republican or imperial period that were tax exempt? Did this change over time?
The Roman state, including the state under the Principate, operated along a more complicated source of revenue collection than simply "taxes" and "not taxes." The modern English term "taxes" fails to describe comprehensively or adequately the realities of Roman revenue collection. Broadly speaking, the Roman state funded itself through the collection of vectigalia (sometimes used in our texts as a catch-all for all sources of state funds) and tributum. You'll find the difference here in the first paragraph. The tribute had originally been levied on citizens (and was originally often thought of as a sort of loan, which the state might repay) for funding distant wars, but from 167 on it was levied on non-citizens only. The extension of citizenship to all Italy meant that after the first third or so of the first century BC the tribute was levied exclusively from provincials. Tribute made up the bulk of the Roman state's revenue, and is the closest thing to the modern concept of "taxes," which are paid to the state annually regardless of an individual's actual commercial activity. As such, in that respect at least, senators were exempt from taxes under the Principate, but so were all citizens. The vectigals, on the other hand, were levied on everyone regardless of their citizenship status, but were duties imposed on certain taxed activities. The use of ager publicus, the importation of goods, etc. were subject to vectigals. If one was not involved in taxed activities one did not pay taxes for them--likely much, if not most, of the citizen population paid no vectigals, and those that did so regularly made up a small minority of traders, landholders, and certain types of farmers (it's unclear whether sharecropping on ager publicus was really a thing). Indeed, senators, equites, and other members of the first (centuriate) voting class regularly paid certain vectigals that the rest of the population likely never paid in their lives, or only rarely. The majority of the vectigals were either leases on ager publicus or portoria, the import taxes, but included in the vectigals were certain other duties unlikely to be paid by the unlanded or by poor farmers. The vicesima libertatis, a 5% tax on the value of a freed slave, had been around since before Augustus' time--while urban shopkeepers and some middling farmers likely had and freed slaves, the bulk of the revenue from this vectigal came from the wealthy. The centesima rerum venalium, a 1% tax on sales by auction, existed in the early Principate, but was reduced and eventually abolished for inhabitants of Italy under Caligula. In 5 AD a lex Julia promulgated by Augustus established the vicesima hereditatum, a 5% tax on inheritance by testament. Testamentary inheritance was fairly common at Rome among most classes that had anything to pass on, but the bulk of this vectigal again came from the wealthy. Augustus in 7 AD imposed the quinta et vicesima venalium mancipiorum, a 4% tax on the sale of slaves. Originally the tax was imposed on the buyer but Nero transferred the tax to the sellers, with, Tacitus tells us, more or less the same result, since the price of slaves was just raised to compensate. All these vectigals, while not unknown to the rest of the population, were likely exacted most heavily from individuals of senatorial and equestrian rank, who participated in such activities most frequently. This, in addition to the leases on ager publicus, which in the Republic at least was said by our texts to be frequently used for the pasturing of the herds and flocks of estates owned by wealthy landholders