I entered my data with several replicates for each X value. Can I perform the runs test following linear or nonlinear regression?

Prism 5 does not try to do the runs test when you entered replicates. Instead, it does the replicates test for lack of fit. 

When you fit your replicate data with nonlinear regression, you can choose whether you want to fit the means or individual replicates. If you fit individual replicates (which often is the best choice for finding the best curve), the results of the runs test from Prism 4 don't really make sense. A run is a series of adjacent points on the same side of the curve. How do you define a series of adjacent points, when several points have the same X value? The entire concept of a runs test is pretty ambiguous in this situation. You should either ignore the runs test results, or repeat the regression choosing to fit the means rather than the individual replicates.

Prism 4.01 and 4.0b and later will do this automatically. Even if you entered replicates, and chose to fit the curve to individual replicates, the runs test will be performed on the mean values.