Is it dangerous to induce before 41 weeks?

Depends. Induction always carries some risk but so does the reason why you are inducing. Induciton should not be done for convenience. It should have medical indications! one must weight the benefits versus the risk with any procedure. Prematurity is the main risk for induciton. Know your dates and know your risks!
Dates might be wrong. Right now, the march of dimes and am college of ob-gyne have a campaign to prevent 'elective" deliveries before 39 weeks. If the due date is off by a few weeks, and is less than 39 instead of 41, then there can be problems related to prematurity. Also, risk of ceasarean if the cervix is induced when not ready, especially if first pregnancy.
Sometimes. Induction does increase the chance of a cesarean delivery that might not have happened if labor started on its own. If the life or health of the mother is at risk, inducing labor is often the right thing to do. But elective inductions (without any medical indication) should be resisted before 41+ weeks.