The leading case is United States  v. Wong Kim Ark.  That case involved parents who were in the U.S. legally when they had a child (i.e. they had "permanent domicile and residence" in the U.S.), so it's not directly on point regarding birthright citizenship for illegal aliens.  However, the Court in Wong Kim Ark gave some big hints.  The Court in Wong Kim Ark stated that:

“The words ‘in the United States, and subject to the jurisdiction thereof,’ in the first sentence of the fourteenth amendment of the constitution, must be presumed to have been understood and intended by the congress which proposed the amendment, and by the legislatures which adopted it, in the same sense in which the like words had been used by Chief Justice Marshall….”

The prior opinion by Chief Justice John Marshall was The Exchange v. McFaddon. Here’s an excerpt from John Marshall’s opinion in that case (emphasis added):

When private individuals of one nation spread themselves through another as business or caprice may direct, mingling indiscriminately with the inhabitants of that other, or when merchant vessels enter for the purposes of trade, it would be obviously inconvenient and dangerous to society, and would subject the laws to continual infraction and the government to degradation, if such individuals or merchants did not owe temporary and local allegiance and were not amenable to the jurisdiction of the country. Nor can the foreign sovereign have any motive for wishing such exemption. His subjects thus passing into foreign countries are not employed by him, nor are they engaged in national pursuits. Consequently there are powerful motives for not exempting persons of this description from the jurisdiction of the country in which they are found, and no one motive for requiring it. The implied license, therefore, under which they enter can never be construed to grant such exemption.

So, it's not 100% clear from this how Marshall (or the Court in Wong Kim Ark) would have approached the issue of birthright citizenship for illegal immigrants.  But, we have two important criteria that Marshall mentioned: (1) continual infractions of the law, and (2) license to enter.  Obviously, illegal immigrants have no license to enter, and granting automatic birthright citizenship is obviously an incentive for continual infraction of the laws.

The Supreme Court precedents specifically say that children of foreign diplomats, and also children of occupying forces, are not subject to the jurisdiction of the United States, and do not automatically get U.S. citizenship upon birth.  I don't think that it would be a huge problem for Congress to treat illegal immigrants like foreign diplomats who are not subject to the jurisdiction of the United States; for example, the individual states would still have full jurisdiction, even though the federal government would not.  In any event, even if Congress does not want to treat illegal immigrants more like foreign diplomats, still the factors described by Chief Justice Marshall suggest that they fall within the 14th Amendment's jurisdictional exception.

I previously made some of these observations in a comment thread, but thought I'd make a full diary post out of them.