The Copyright Clause and the First Amendment
There is a tension between the First Amendment and the Copyright Clause. The First Amendment to the Constitution reserves several freedoms for the people, one of which is the Freedom of Speech.  But copyright effectively makes some speech illegal by giving the copyright owner power to prevent another person from using particular words or expressions of creativity. The Supreme Court has said that this limitation does not violate the First Amendment, because copyright promotes speech: "the Framers intended copyright itself to be the engine of free expression," and, through the Copyright Clause, gave Congress the power to establish a limited monopoly over certain expressions.  Thus, there is a delicate interplay between the need for free speech and the need to compensate the speakers. "In establishing this balance 'on the copyright side, economic encouragement for creators must be preserved and the privacy of unpublished works recognized. Freedom of speech[, on the other hand,] requires the preservation of a meaningful public or democratic dialogue, as well as the uses of speech as a safety valve against violent acts, and as an end in itself.' " 
Several doctrines exist to ensure that Copyright's purpose "to promote the progress of science and useful arts," is properly reconciled with First Amendment considerations. The idea/expression dichotomy, the doctrine of fair use, and the limited times requirement all act to preserve the delicate balance between expression and control.
One doctrine that saves copyright from First Amendment scrutiny is the idea/expression dichotomy.  An author cannot receive a copyright in facts and ideas, only in creative expression.  This serves First Amendment goals by ensuring that copyright can't stop people from speaking about information or ideas.
Fair Use Doctrine
Fair use is a statutorily defined set of activities that do not infringe on copyrights; it includes some copying for purposes of criticism and parody.  While the courts have not strongly associated the fair use doctrine with First Amendment principles, many scholars agree that without fair use, copyright would be in direct conflict with the First Amendment.  Yochai Benkler, for one, discusses how fair use allows one to engage in free expression; one is not free to speak when under threat of a lawsuit.  Fair use is discussed in more detail above.
Limited Times Requirement
The limited times requirement proceeds directly from the Constitutional text itself. While there is some dispute as to the extent of Congress' power in this area, it is well settled that perpetual copyrights would be unconstitutional.  Many scholars argue that the limited times requirement reinforces First Amendment rights by placing a limit on the amount of control that can be placed on an element of expression. Eventually, that expression must enter the public domain and be useable by everyone.
In 1997, Congress passed, and President Clinton signed, the Sonny Bono Copyright Term Extension Act (CTEA).  This law extended the terms of present and future copyrights by twenty years. Some say that the CTEA violates the limited times requirement, since Congress could just pass another CTEA and extend copyrights yet again. This is essentially "perpetual copyrights on the installment plan."  A group of plaintiffs including Internet publishers brought suit to declare the CTEA unconstitutional, saying that it violated both the limited times requirement and the First Amendment.  The Supreme Court should rule on this matter in late 2002, their decision may explain more about the legally uncertain relation between copyright term length and the First Amendment.