Is it okay to keep a mobile phone connected to its charger, 24 hours a day, 7 days a week?
Most modern phones have a charging control circuit which takes care of what you are proposing. When the battery is charged, the charge control IC will terminate charging of the battery, then monitor to see if it needs charging again. So all the charger has to do is make the power available. 

Below is a flowchart from the datasheet of MCP73831 Li-Ion charge controller IC from Microchip:



You can see the various stages of the charge cycle. This is a pretty basic IC (I just picked it since I've used it a few times) but does the job and is nice and cheap - many phones will have a pretty complex power management IC, possibly a custom one.