Inner tubes will stretch to fit a bigger tyre, so it is likely to be ok. There is some risk that the stretching will make the tube wall a little thinner, and it can be more prone to punctures, or it can expose a weakness and cause a flat.
Personally I would change the tubes at the same time, as tubes are cheap. You could likely keep the old tubes with the old tyres in case you wanted to go back to the smaller tyres.
Another issue may be clearance of the frame. The bike might not accomodate the wider tyre. Some times the rear wheel will not go into the dropouts properly when the tyre is inflated. That is not fatal, just perhaps annoying that the tyre may need to be deflated when putting the wheel in. Makes it difficult when fixing flats out on the road.