Are there any experiment reproducibility databases? Quite often when I read a medical research article I wonder whether other researchers tried to reproduce the experiment. I could go through the list of papers that cite the article but I find this solution a bit tedious and inefficient for this task.

Is there any database that lists reproduction attempts (with ideally the number of fail and successful reproductions)?
While not a direct answer, publishing failure to reproduce can be difficult as a standalone activity (think cost, time, and rewards), and where reproducibility information does exist, it may be field or community specific (like blog-syn for organic chemistry). 

But there are a few large and well-funded studies that aggregate information simply because their purpose was to replicate big studies- eg the Cancer Biology Reproducibility initiative, and perhaps some of these studies will grow over time.

You may also want to check sites such as Pubmed Commons and PubPeer, which increasingly are gathering discussion of papers.