I concede to London Ed that it is not clear what exactly the thesis of presentism is. There is no point in considering objections to it until we are sure what the thesis comes to. The rough idea is of course easy to convey: only temporally present items exist. This is more plausible under restriction to items 'in time' where the eternal God and abstracta such as Fregean propositions are not 'in time.' The rough idea, then, is that only present contingent concreta exist. This implies that a wholly past contingent concretum such as Socrates does not exist.
But how are we to take 'exist' in the last two sentences? As present-tensed? Then both sentences are trivially true. Surely no philosopher who calls himself a presentist intends
Tautological Presentism: Only present contingent concreta exist at present.
And of course he doesn't intend
Timeless Presentism: Only present contingent concreta exist timelessly.
For that implies that if x is a timeless contingent concretum, then x is temporally present.
But the clear-headed presentist must also, in his formulation of his thesis, avoid giving aid and
comfort to the absurdity that could be called 'solipsism of the present moment.' (I borrow the phrase from Bertrand Russell, Human Knowledge: Its Scope and Limits, Simon and Schuster 1948, p. 181.) To wit,
SPM Presentism: Only what exists (present tense) exists simpliciter; nothing existed and nothing will exist.
The idea behind (SPM) is decidedly counterintuitive if not insane. To illustrate, consider James Dean who died on September 30th, 1955. It is a Moorean fact that Dean existed but no longer exists. (Alter the example to Dean's car if you hold to the immortality of the soul.) It is a Moorean fact that there actually was this actor, that he was not a mere possibility or a fictional being or nothing at all. But this plain fact is incompatible with SPM-Presentism.
Another possibility is
Disjunctive Presentism: Only present contingent concreta existed or exist or will exist.
Disjunctive Presentism seems objectionable because, e.g., Scollay Square existed, but does not now.
Tenseless Presentism: Only present contingent concreta tenselessly exist.
Now the problem is to explain that 'tenselessly exist' means if it does not mean 'timelessly exist' or 'did exist, does exist, or will exist.'
Ned Markosian defines presentism as the view that, "necessarily, it is always true that only present objects exist." (here, fn 1) This is not helpful since we are not told how to read 'exist.' The Triviality Objection threatens to kick in. And how are we to understand, "it is always true"? If this involves quantifying over times, then anti-presentism is let in through the back door. If there is a manifold of equally real/existent times, then presentism cannot be true of these times.