If you wear a platinum wedding ring and a white gold engagement ring will they go different colours over time?

Yes, they will. White gold is not naturally white like platinum--it is alloyed with other metals and plated with rhodium to give it it's white appearance. The plating on a white gold ring will eventually wear away, and you will need to re-plate your ring to restore it's white appearance, or else it will start to take on a yellowish tint.

You should also know that a platinum ring will accelerate the wear on a white gold ring because platinum is a stronger and more durable metal than white gold. The platinum ring will constantly be rubbing up against the white gold ring if they are both worn on the same finger (which is the traditional way of wearing the rings), and this friction will wear away at the plating of the white gold ring.

For these reasons, it's preferable to get matching metals for your engagement ring and wedding band.