DVD discs will never damage a player.
DVD and CD technology let the lens read the disc but there will be no contact between the lens and the disc. Also discs will never be scratched by the player because there's no contact between disc and tray either (you will notice that on the most common tray type, used on standard DVD players, the DVD disc perimeter will be the only point of contact with the tray and there's no data on the 3 mm outside perimeter of the disc).
Discs could actually get scratched when there's no proper handling (when taking the disc off/on the case and into the player's tray or when cleaning the disc).
Lens would not read discs when the lens itself or disc are dirty or scratched somehow. Also a specific player could not recognize a specific disc if it's recorded with a not supported format. Some discs are just damaged (bad recorded that won't play even when not scratched or dirty) but this usually happens when they are home made copies or illegal ones. I don't know if it could actually happen to brand new DVDs but for you to be sure just go to the store and get it tested there. Nevertheless there's no way this disc damage the player.
But it's possible by using laser lens cleaner to scratch the lens. Some manufacturers do not recomend using these.
I'm a DVD movie collector and have all my discs unscratched. I've never used lens cleaner either. So my recommendations are:
The discs:
Be careful taking the disc from/into the case.
Protect the discs from dust avoiding turning them bright-side up and never touch them on that side to prevent fingertips. That way you'll minimize the need to cleaning them.
Handle the discs only from case to tray / tray to case. Under no circumstances leave a disc exposed.
If dust needs to be removed from disc use a very soft lint-free cleaning cloth and wipe it very gently in radial (non circular) moves only.
Be careful taking the disc from/into the player's tray.
The player:
Protect the player from dust by using a cover so no lens cleaner will be needed.
But there are two basic types of player trays out there:
1) Players where only the tray goes out (the lens are kept inside the player's case) then you just put the disc on the tray. This kind is used in almost all comercial DVD players and desktop computer's DVD player drives. This is used too on the first XBOX game console.
These are the most recommended type of tray. They will never scratch discs and lens is never exposed.
2) Players where the tray goes out with lens incorporated (or a cover opens up) and then the disc is attached to a spining wheel at the center of the tray. (This kind is better used in Notebooks, portable DVD/CD players and some game consoles like dreamcast and playstation 1/2). I don't recommend this type of players. The lens is totally exposed so it will accumulate dust easily with increasing disc reading errors and discs are also scratched easily when manipulated in/off the tray.
As an additional note: There are also the self-loading no-tray type like the ones used on car audio CD/DVD players and game consoles like nintendo wii and playstation 3. I will never recommend buying a player with this type of tray.
If player stop reading discs that it usually had no problem to play (like yours) then it's better to take it to qualified technical service. Avoid using lens cleaners yourself.