Thanks for your message! You may want to look at
Janet's answers and
Ziheng's answers to two similar questions asked some years ago -- there may be other similar questions asked in this google group, but these are the most relevant ones I found when searching for "not significant LRT".
Following the answers given in the other thread of messages, you may want to run CODEML a few times to confirm the results are stable. Please note that CODEML will always print the BEB results regardless of the results you get with the LRT . Nevertheless, if the LRT is still not significant (i.e., you fail to reject the null hypothesis, which in this analysis means that there is no statistical evidence for positive selection), you would not proceed to search for sites under positive selection as you are not supporting evidence for that case. If you have very high posterior densities (i.e., 95-99%) in the BEB results, perhaps there is weak evidence for positive selection, and that's why the BEB results conflict with the LRT (see
Ziheng's answer in this thread).