Since Kakashi's sharingan is technically Obito's, would he go blind if Obito used his sharingan too much? Or are they now separate since they both have one eye each.
I would argue that he wouldn't.
First of all, we can see Kakashi turning weak every time he used Kamui, although Obito uses it as much as he wants and seems to be alright each time.
Now, if Obito using the Mangekyou were to weaken Kakashi, then we could conclude that Obito overusing his Mangekyou would blind Kakashi. However, nothing happens to Kakashi while Obito uses Kamui. The only connection between the two eyes apparently is the dimension that the Kamui transports to.