I have strong publication record in my Ph.D. and 1st Postdoc that went for two years. Got 4 TT interview calls before last year.. but not successful in any. Last year did not apply, but this year, after fixing a few shortcomings noticed in my previous interviews, started the search again. In the meantime I published a couple of top journals as the corresponding author. However, I am not in harmonious professional relation with the present postdoc advisor. I am working for a little over one year in this position but did not yield any journal publication. I am sure that this advisor's reference will have negative impact on my search, if at all he agrees to write one. The only best way for me is avoid this advisor's letter, since I can arrange the required number of good letters from my previous affiliations. If I proceed with out my present advisor's reference, will it have any bearing on my chances? Expecting some suggestions from the people having experience in search committees.
In my view, strong letters are extremely important. You certainly don't want any letters from people interested in sabotaging you in any way. Even poorly written letters can be a setback. So can letters written by non-native speakers who may not grok the nuances of certain phrases.
There is the concept of Damning with Faint Praise that can be absolutely deadly in a letter. It can be intentional or not, actually.
Get letters from people who know your potential and are willing to speak for it. They need to know about your past work, of course, but people will be looking for potential and how it relates to a particular position.
If you have any doubts about what a person might say, ask them about it, or go elsewhere. In certain situations (non-native speakers, say) you might even request that a letter writer check their letter with a neutral third party, such as a department head. This would have saved me tremendous grief long ago.
There is no problem in asking other people than your postdoc host to provide letters for you. It is you who decide who to ask, and recruitment committees will probably not even notice, nor will care whether your postdoc host is missing (unless there exists a special relation between a committee member and the host, in which case you can still not ask the host for a letter).
I recommend having someone from your current institution write a letter showing that you are good to work with.
I have served on several search committees. I read letters not for information on research productivity (which is better shown in the CV) but for indications of how the candidate would be as a co-worker. I do not want to hire a jerk.
For someone in your situation, I would notice that you did not have a current reference and I would be concerned that you might be a jerk. If a co-worker, preferably someone who is more senior than you, can speak to what it is like to work with you and perhaps even show that your advisor is the problem, that letter would make me much less concerned.