I am accessing the twitter streaming API. I generate a map using Basemap in python. 
I want only certain parts of the map to change with time (for eg. every second). Is it hard to do? 
Do I need to leave Basemap and look for something else? Please help!
A possible approach: divide the map into tiles, and treat each one separately; use Basemap to generate just the map-tile that contains new data, then update just that tile on your webpage using Ajax. 
Of course, depending on the nature of changes to the data on your map, this approach may or may not work for you -- gerrymandering is not really possible. 
You would need to write logic to understand which tile the new data belongs to, then use basemap to create a new image for that time, then intelligently update the tiled image. You will also have to play with margins and padding (both in matplotlib and in CSS) to cleanly piece the tiles together. 
...
When the approach gets this complicated, one should re-evaluate whether better tools are available. Basemap doesn't sound like a good fit for what you need to do.
Dynamically updating images on website