I see that the soldering iron I am looking at uses tips that Weller only sells pre-tinned with lead free solder. I will NOT be using lead-free solder, I will be sing plain old 60/40 solder.

Will this cause any problem or unfavourable condition whatsoever for the soldering work or the tip life/effectiveness?

I'd assume there wouldn't be a problem. Either way, tips are cheap, and you could always sand the tip if the solder doesn't stick. It wouldn't affect later soldering, just the first tinning.
It'll be no problem. Lead free tips are designed to handle the higher melting temperatures of lead free solder. You will quickly wear out a tip meant for leaded solder if you use it with lead-free solder.