Learn English – Any website to look up words with similar pronunciation or spelling

I don't think there are hard-and-fast rules when it comes to words that have been recently assimilated into English. The pronunciation that usually sticks is a halfway house between the original foreign pronunciation and what is easy for Anglophones to actually say.

To take your two examples:

A word like tortilla can just be pronounced as if it were an English word, tortia, without any Spanish accent. But you will still find English speakers who pronounce the double l, just like the do with paella and Marbella, and also the j of fajita and jalapeno as [dʒ], since they are unaware of the Spanish pronunciation.

The sounds in the Japanese word karaoke are a bit more foreign. The r in there is a very different liquid to the English r, English speakers aren't used to gliding from [a] to [o], and English words seldom end with with an [e]. Hence the slightly mangled /ˌkæriːˈoʊkiː/

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 _".