For many arguments of natural theology, this question is simply irrelevant. However, the kalam cosmological argument (KCA) requires that universe began to exist at some finite time in the past:
1. Whatever begins to exist has a cause.
2. The universe began to exist.
3. Therefore, the universe has a cause.
In support of (2), Bonaventure reasoned that the present would not have arrived if the past were infinite. Craig summarizes this particular argument as follows :
2A. The temporal series of events is a collection formed by successive addition.
2B. A collection formed by successive addition cannot be an actual infinite.
2C. Therefore the temporal series of events cannot be an actual infinite.
(2A) assumes an A-theory, or dynamic theory, of time. (The other philosophical argument that Craig defends - i.e. the impossibility of an actually infinite number of things - does not necessarily assume an A-theory of time.)
(2B) is supported by the fact that no matter how many members are added to a set, it is always and indefinitely possible to add another before arriving at infinity. Hence, any collection formed by successive addition will always be finite.
While Thomas Aquinas agreed that the universe began to exist, he disagreed with Bonaventure that the universe's beginning could be demonstrated by reason alone. Thomas reasoned that the universe's past, while not actually infinite, could be potentially infinite.
Thomas, I think, is technically correct. However, in order for the past to be potentially infinite, the past would also have to be growing. In other words, time would have to be moving backwards, and not forward only. Since this is contrary to experience, the objection that Thomas offers is generally not taken as a very serious threat to the KCA.
Another objection states that an infinite collection can be formed by successive addition if the collection has always been being formed. Any moment in the past is finite, so if there are infinitely-many of them, the collection as a whole will be an infinite set. This objection appears a bit fishy to me, and I don't just mean that it appears to be question-begging (e.g. the past is infinite if it is infinite). It is true that at any time in the past, there is a finite distance from that moment of time to the present. The problem is that even though each finite period of time may be traversed, it doesn't follow that the infinite set as a whole could be traversed. In short, the objection is susceptible to a composition fallacy.
Obviously, entire books can and have been written on this one argument alone. However, it seems to me that the objections put forward so far are unsuccessful. Barring any more cogent objections, it appears that it is reasonable to suppose that the universe cannot be infinitely-old on an A-theory of time.
 William Lane Craig, The Kalam Cosmological Argument, Wipf and Stock Publishers, 1979, p. 103. I have changed the premise numbers/letters in order to avoid confusion with the general KCA summary.