There is no evidence that I know suggesting that knowledge of ancient Greece and Rome had much influence on the thinking of the "founding fathers," but of course knowledge of history will always influence political thinking at some level. For example, in Britain the upper house of parliament was (and is) called the "House of Lords"--not a viable name for the corresponding American institution, so borrowing the term from Rome, they named it the "Senate".
Some indications of ideas from the ancient world that informed the founding fathers are in the Federalist Papers, a summary of which points are listed below:
In Federalist No. 6 it reads:
Sparta, Athens, Rome, and Carthage were all republics; two of them,
Athens and Carthage, of the commercial kind. Yet were they as often
engaged in wars, offensive and defensive, as the neighboring
monarchies of the same times. Sparta was little better than a
wellregulated camp; and Rome was never sated of carnage and conquest.
Federalist No. 18 ("The Insufficiency of the Present Confederation to Preserve the Union") is a long summary of the Achaean League and politics of ancient Greece . The general point was to argue that confederacies (like the colonial states) were weak due to dissension. This was an argument for Federalism (the introduction of a central power).
As a weak government, when not at war, is ever agitated by internal
dissentions, so these never fail to bring on fresh calamities from
abroad. -- Federalist No. 18
In Federalist No. 34, the example of Rome is cited to demonstrate that a bicameral Congress can be successful:
It will be readily understood that I allude to the COMITIA CENTURIATA
and the COMITIA TRIBUTA. The former, in which the people voted by
centuries, was so arranged as to give a superiority to the patrician
interest; in the latter, in which numbers prevailed, the plebian
interest had an entire predominancy. And yet these two legislatures
coexisted for ages, and the Roman republic attained to the utmost
height of human greatness.
In Federalist No. 41 Madison used the example of Rome to argue that the existence of standing armies should be limited due to the threat they might have to liberty:
The veteran legions of Rome were an overmatch for the undisciplined
valor of all other nations and rendered her the mistress of the world.
Not the less true is it, that the liberties of Rome proved the final
victim to her military triumphs; and that the liberties of Europe, as
far as they ever existed, have, with few exceptions, been the price of
her military establishments. A standing force, therefore, is a
dangerous, at the same time that it may be a necessary, provision.
In Federalist No. 63, Rome is cited as a justification for creating a senate:
It adds no small weight to all these considerations, to recollect
that history informs us of no long-lived republic which had not a
senate. Sparta, Rome, and Carthage are, in fact, the only states to
whom that character can be applied.
Later in the same essay Madison cites ancient examples of representative democracy and how the senate will not be a threat to it:
Lastly, in Sparta we meet with the Ephori, and in Rome with the
Tribunes; two bodies, small indeed in numbers, but annually ELECTED
BY THE WHOLE BODY OF THE PEOPLE, and considered as the
REPRESENTATIVES of the people, almost in their PLENIPOTENTIARY
capacity. The Cosmi of Crete were also annually ELECTED BY THE
PEOPLE, and have been considered by some authors as an institution
analogous to those of Sparta and Rome, with this difference only,
that in the election of that representative body the right of
suffrage was communicated to a part only of the people. From these
facts, to which many others might be added, it is clear that the
principle of representation was neither unknown to the ancients nor
wholly overlooked in their political constitutions.
... The Tribunes of Rome, who were the representatives of the people,
prevailed, it is well known, in almost every contest with the senate
for life, and in the end gained the most complete triumph over it...
It proves the irresistible force possessed by that branch of a free
government, which has the people on its side.
Hamilton used the examples of Rome in several instances to argue for a strong single executive (rather than multiple, Consul-like executives):
The Decemvirs of Rome, whose name denotes their number, 3 were more to
be dreaded in their usurpation than any ONE of them would have been.
In general, the different theorists tended to cite the example of Rome and Greece more to bolster individual arguments than to serve as a model. However, in a few respects, such as the institution of the Senate (and its name) is due to the inspiration of Rome. Also, the idea of having two senators from each state may have been inspired by Rome having two Consuls, although it does not explicitly say this in the Federalist papers.