Why do universities place a weight on GRE/TOEFL scores?

GRE/TOEFL scores are used in a number of different ways (some of which are alluded to above)

• GRE scores are sometimes used as a university-level filter (if your GRE score is < X, then you'll need a strong support letter from your department to get admitted)
• TOEFL scores are used as a filter for giving people TA funding (if your TOEFL score is too low, you can't be assigned to be a TA, and if it's lower, you can't even get RA funding)
• More informally, the GRE/TOEFL scores are used as a "do you even care' filter: for a CS program, a quantitative score less than 600 might be considered to be a warning that the candidate doesn't even care enough to prep for it.

But for Ph.D programs, the GRE/TOEFL are either used as a high-bar disqualification filters to prune applications (in top schools), or as low-bar disqualification filters to prune the non-serious applications.

On the admissions committees I've been on, GRE scores have been used primarily for one purpose, namely dealing with students from out of the way places. Every year, we see a number of applications from students at not very prestigious schools but with perfect grades and letters saying they are the best in years. We suspect that the courses are easy and the competition is not impressive (being the best is meaningless if we doubt the second best is very good), but we don't want to reject someone unfairly. GRE scores give a simple, consistent way to compare these students with those from other schools. Most of them have unimpressive GRE scores, but occasionally they do very well on the GRE, and in those cases we investigate further.

TOEFL scores are another case in which consistency is very helpful. Skype interviews would give more information, but different interviewers would be more or less demanding (plus the interviews would be a lot of work). If you want to set a consistent cut-off, for example for TA support, then a standardized exam may be the right approach. Note also that the administration may not trust faculty interviewers not to exaggerate the English abilities of students they want to admit.

If it weren't for its usefulness in screening the applications from out of the way places, I'd be in favor of eliminating the GRE entirely. However, even making someone jump through a meaningless hoop can actually be a useful filter. In practice, having a successful career requires occasionally doing things you don't care about, for reasons that may not be clearly explained (and might or might not turn out to be justified if they were explained). Some students run into serious psychological issues here. Maybe they can't overcome their disorganization if they don't feel motivated, or maybe they just refuse to participate in anything without a clear justification. It may be unfair, but these students are not likely to be successful in the long run, and it's a waste of time and energy to prepare them for a career that isn't likely to work out. Jumping through hoops like the GRE is a mild test in this direction. If you are too disorganized to sign up for it in time, or if you aren't willing to jump through such a hoop at all, then that's a bad sign for your future career.

My (computer science) department does not require GRE scores at all. A few faculty still use them to evaluate applicants—there's no accounting for taste—but there is certainly no official cutoff (as Suresh suggests). In practice, they're only useful if they're really low.

However, by state law, an international student at my university cannot be hired as a teaching assistant if their TOEFL spoken English score is below 24. (Too many courses were being taught by foreign students with thick accents.) Since my PhD program has a TAship requirement, we must filter out applicants with low TOEFL scores. We can still admit applicants with scores below 24 if someone offers them an RAship, but they have to bring their score up to 24 by the end of their first year. So in practice, you have to have at least a 20 to be considered at all. (And we do conduct phone interviews with borderline cases.)