The OP surely has solved this problem by now, but I found this question when faced with the same problem, so the answer will probably be helpful to somebody.
At the time of this writing the current version of Eagle is v8.3.2; the procedure may be different in other versions.
To delete a dimension, just choose the trash icon and then click somewhere in the "hot zone" for the dimension marker. Where is the hot zone? Ah, there's the rub: it is unmarked and very tiny. Random clicking in likely locations will generally miss it. Fortunately, though small and invisible, its size and position are predictable.
The hot zone is 1 mm tall and 1 mm wide wide. In the versions of Eagle I have used it is found positioned at the exact midpoint of the dimension line. An additional hot zone is found at the spot where the user clicked to begin creating the dimension, but this hot spot is less easily located.
The OP has also encountered a second problem that is very common: if you naively add a measurement (dimension) line, it will by default likely be placed on either the top or bottom layer. This is a bad idea because the lines will be treated as traces that may trigger DRC rules. The OP encountered this when the DRC complained about trace overlap errors.
AutoDesk recommends placing measurements on layer 47 (Measures). If you do so, you will not trigger DRC rules.
There is another common source of confusion when measurements are placed on the Top or Bottom layer: the default settings violate common DRC rules for line width and for font, so the DRC creates a hatched design covering the lines, arrowheads, and the text of the measurement. Even if the actual dimension is then successfully deleted, the hatched coverage remains, making it appear as if the dimension line and arrows still remain. This is the problem that user1155386 has noted in the comments above.
It is not possible to delete this phantom dimension line, but it will go away if the DRC is run again or if the file is closed and reopened.
I hope this answer will help somebody, because personally I found these problems somewhat frustrating and did not find any answers in the documentation or in other questions on this site or other sites.