I just bought a 5 string bass and was about to tune it to EADGC, but I really don’t know what I am doing so I thought I would ask first. I have read that most tune it to BEADG, which is what I have it at now, but I would like to know if it would hurt the guitar over time if I tuned it to EADGC? Could it warp the neck?
I always tuned mine to BEADG, mostly because I liked the sound the low B-string made, and it simply never occurred to me to try raising it up a fourth. It really depends on the sizes of the strings you have, but most 5-string sets are intended for tuning beginning at the low B. If you want EADGC, buy a 6-string set and just don’t use the B-string. That could get expensive, however.
You’re going to waste a perfectly good string on a C!?!?
Remember, you’re playing a BASS, not a guitar. 
I think basses/guitars are tolerant of different tensions. If you are still worried, get lighter guage strings, which won’t be as tense to get the same notes.