In a word—yes, drain cleaners can damage your plumbing, or at least they have the clear potential to do so. After all, they are highly acidic in nature, with caustic and oxidising active ingredients. You don't have to refrain from using them altogether, but the trick is to use them wisely so that you can avoid expensive damage to your plumbing. Essentially, they should be a last resort.
Prevention Is Better Than Cure
There are a number of (largely) natural methods that claim to unblock drains. They might be effective for minor blockages, but they're not going to pack the punch of a chemical drain cleaner. Baking soda and vinegar can work when it comes to a partial blockage, but again, it's not going to work when your pipes are completely clogged. Some of these cleaning methods are more appropriate as a way of flushing out your drains, which can prevent a major blockage from occurring. Think of them as a form of prevention. Flushing your drains with boiling water from an electric kettle is about the easiest way to flush out your drains, thus minimising the chance that you will have to use a chemical drain cleaner.
A More Natural Approach
You might also want to try a natural product designed to clear drains. Look for a cleaning product that contains bacterial enzymes. These products are usually sold as granules which are sprinkled into the drain, releasing harmless bacteria which then consume any organic matter inside your pipes. The effects are not as immediate as a chemical drain cleaner, but as the bacteria only consume organic material, your pipes will be left totally untouched. These bacterial enzymes do not cause damage to the environment and only have a short period of activity, meaning that they will not continue to consume organic material once they have been flushed into the sewers.
Using Chemical Drain Cleaners
Perhaps there's a blockage that one of the more natural methods will not be able to handle. This can be true if the blockage is caused by anything non-organic. So yes, in some instances a chemical drain cleaner can be your best bet. Use them sparingly and obtain a product that is specifically designed for the types of pipes you have in your home. If you have plastic pipes as a component of your plumbing, then do not use something specifically designed for metal. Use the product sparingly, and don't be afraid to use less than is recommended before seeing if the product has been successful.
Sometimes it can be necessary to use a chemical drain cleaner, but in order to preserve your pipes, they should not be your first port of call. For really tough clogs, consider contacting a professional drain cleaning service for assistance.