Is it dangerous to teach yourself to sing without a teacher
Its certainly not 'dangerous', but it's not very effective. I myself had seen many books and videos and DVDs all over about teaching yourself to sing, but none of these can actually give you feedback from any of them. Sometimes your FRIENDS or your spouse can tell you that you sound great or not, but they wont be able to point out the mistakes and hence will not be able to give any deep rooted constructive feedback.
If you aspire to be professional singer then you must get a good and reputed singing teacher. If you just want to get better at singing then you can take some online lessons and keep improving. Sometimes after spending hours on online lessons its actually frustrating if you dont see any good results.
Having said that some techniques shown in videos may be harmful if not done correctly.
Some people who run singing institutions will always advice you NOT to learn anything online. However I would say do watch the videos and see what works for you and what helps you. There are many aspects of singing like your ability to clearly articulate the words, sing without shyness, singing with experessions, some of these aspects can be corrected and having some basic knowledge always helps.