The 10 speed chain is slightly narrower than a 9-speed chain. The tooth spacing of 9 speed chainrings in the side direction is slightly different than for 10 speed chainrings. In the front-back direction, the tooth spacing of all chainrings is the same regardless of the speed count.
The worst that could happen (assuming here you have at least two chainrings) is that when shifting in the front, the chain starts to momentarily skate on the small ring teeth. This phenomenon is most likely to happen when using top gear (large front chainring, small rear sprocket) and downshifting in the front to a smaller ring, before shifting in the back. This shifting pattern results in an extreme chain angle and is not reasonable for this reason. In practice, you will never shift this way, so the problem should not occur with reasonable shifting patterns.
This effect will not ruin your chain. It's made of steel and thus very strong. Your cassette has no contact with the chainring, so it will naturally not be ruined.
In practice, when having at most 1 speed mismatch on the chainrings, it is extremely unlikely that the problem would occur. A larger problem might occur if using 8 speed chainrings with 10 speed chain.