Why don't economists do regression diagnostics? There is a lot of talk about regression diagnostics in tutorials on the web, but then in economics research papers nobody actually reports residual plots, collinearity checks etc. Is there any reason for this?
To start with the question in the title:

Why don't economists do regression diagnostics?

People do regression diagnostics. I don't know of any respectable researcher that would not perform regression diagnostics and virtually any paper will have hints that the regression diagnostics was performed. For example, in tables with regression results you will find remarks on White or HAC errors being used to either correct for heteroskedasticity or autocorrelation or both, or references to corrections for cross-sectional dependence etc.
Only very unscrupulous scholars would make claims about these issues being/not being present without some testing.  So scientists (or at least the good ones) always perform regression diagnostics.
To address the question in the body:

in economics research papers nobody actually reports residual plots, collinearity checks etc. Is there any reason for this?

Yes, almost all scientific journals have very strict page limit of between 30-60 pages with most journals having page limit around 40 pages. In addition shorter articles are often more preferred and attract wider readership because people usually prefer to read shorter papers. Also, note that page limits are usually inclusive of list of references which often can eat up another 1-5 pages and also all other stuff. Only online appendices are excluded from page limit.
Now documentation of regression diagnostic can easily eat up 10 pages if you want to do it properly with all the plots (or even more). Moreover, regression diagnostic is not of much interest in itself. You preform it in order to know how to properly specify your model or what identification strategy to use. Once you figure that you just use the appropriate model, so by itself regression diagnostic has little value for reader as it carries very little information about research result. As mentioned in the first part of the answers people will still mention in their paper that there was autocorrelation or heteroskedasticity and how they corrected for it (and so on for other problems), so there is not that much point in additionally wasting space in the paper on showcasing all the auxiliary diagnostics. Any mistrustful researchers can just request their data and rerun that diagnostic themselves.
Consequently, the reason for that is simply that there is not enough space for it in the paper, and because you always have to economize on the space given. If you would really want to do it, it would usually end up eating 1/3 of a precious space, You would be surprised how common problem it is for researchers to actually fit their research in the page limit. Often you will be forced to relegate even main derivations to online appendices just to fit in the limit. In the end anything that is not of great importance to support or interpret the main result will simply not make the cut.
As mentioned in the comments this can sometimes cause issues with reproducibility, but nowadays the solution to that is that journals require scholars to post their code that was used to derive results (where you would find also regression diagnostics) rather than actually report it in the paper, for the reasons mentioned above.