This is to a large extent a question of how reliable the data in the database needs to be. Reliability (and spatial scale) will differ between datasets and between species groups within datasets, and it is difficult to give a general recommendation. I doubt that you will find a single database with good coverage over all taxonomic groups, even if it is in the form of country checklists. For the most reliable information, curated country checklists for specific taxonomic groups will probably be best, but these have to be searched for individually for each taxonomic group of interest.
As a starting point, you might want to look at the occurence data that can be found in gbif.org (The Global Biodiversity Facility). The data found there is certainly not complete, and it will be misleading for many species. However, for the current distribution of relatively well-known groups of species it will give you a good idea of their distribution. This has to be evaluated on a case-by-case basis though. You can access the data in gbif using external tools, for instance using R through rgbif (there is also tools for python or other languages). At the blog recology.info you can find a tutorial on how to get a species list for a particular country using rgbif (more specifically the function