I've just bought 23mm wide tyres to replace my current 25mm ones. Do I need to buy 23mm tubes too?
What you want to avoid is:
a tube which, when deflated, or slightly inflated to a torus shape is already as wide as the inner space in the tire or wider. The tube should be quite a bit smaller than the inner tire chamber when slightly inflated.
a tube with an excessive diameter: a situation when you slightly inflate the tube to insert it into the tire, and it is too large, so that it has to be awkwardly stuffed in there.
a tube with too small a diameter which basically stretches onto the rim. It is possible, but installation is difficult.
If a tube has a slightly small radius, but pre-inflating it expands the radius sufficiently that it can be comfortably stuffed into the tire, and the required inflation doesn't make the tube fatter than the tire chamber (the tire clinches onto the rim easily, and the tube doesn't get in the way), it is probably okay.
a tube which is too thin relative to the tire, and has to stretch ridiculously. I use 700x(35-43)c in 45mm tires; I wouldn't want to go much narrower.
As far as the numbers on the box go, you can be slightly out of the tube's range. There is no such thing as a 23 mm tube. There are tubes which include 23 mm in their range. Your existing tubes in your 25 mm tires could already extend over the 23 mm range. Suppose your existing tubes are "22-25". Then, the 23 mm tires are in the middle of that range. Or suppose your tubes are 25-28. Then the 23 mm size is a bit out of range: it's okay to re-use the tubes if they pass the above avoid guidelines.
Generally inner tubes will fit a range of tyre sizes, for example 18-28mm. Because both 23mm and 25mm are standard road tyre sizes and close together, you're unlikely to have an inner tube that won't fit 23mm.
Ideally if you can identify the inner tube, you can check on the manufacturers or retailers website.
Failing that, give it a go. Fit the inner tube, and tyre and inflate as normal. If the inner tube is too fat, it will force the tyre away from the rim as it inflates, so after inflation check round the rim on both sides that the tyre is seated properly.