Online tools to look up words from pronunciation

Are there online tools to look up words from their pronunciation? For example, when I hear /bεə/, I want it to tell me that there are words bear and bare that are pronounced /bεə/.

I do not know of any online tools for doing reverse pronunciation to word search. However, if you are handy with searching text files, you can download the CMU Pronouncing Dictionary. It does not use IPA, but there is a key to the phonetic encoding that dictionary uses (called ARPAbet). You can then search the text file for pronunciations you are interested in. To take your example, /bεə/ in ARPAbet would be B EH1 R so a search for that returns these results:

BAEHR(1)  B EH1 R
BAER  B EH1 R
BAHR  B EH1 R
BAIR  B EH1 R
BARE  B EH1 R
BEAR  B EH1 R
BEHR  B EH1 R

Note the dictionary is not 100% perfect or consistent, as it was generated by low-paid grad students, but it is freely-available. It has many proper nouns not specially marked, so you'll need to look up results elsewhere to see what they mean if they are unfamiliar. Also, the dictionary is American English only, so if you're hoping for British pronunciations, you're out of luck there too.

But you can definitely do what you want, and find answers to questions like "all words that sound like _".