Effect for browser:
Though this looks like a bit of work for web browser, but technically it does not make much of a difference. The browsers are too fast to handle these relative url structure and make a call to application server
Effect for application Server:
None, as it needs to return the requested file (relative/absolute link ultimately maps to a web path)
Effect on page size:
Yes there would be some reduction in size (again not something that would make a huge difference to your page's performance that could be achieved by something like content encoding gzip or minifying resource)
So i think technically the absolute/relative urls dont make much of a difference on page speed / any weightable matrix.
Yes it make huge difference in managing multiple environments like dev, pp, prodpp etc
Example :
on your local development you might have dev.example.com
on pre production you might have : pp.example.com
.
.
So in those scenarios it would be relatively easy to manage code with relative urls (though can be managed by environment settings also)