It is not mandatory to sell. As explained above, you have two options, either to apply for rights shares or sell them in the market if you do not wish to apply. If you do not apply for rights shares, the REs will expire worthless.
Though there is no loss for you as the REs are alloted at no extra cost. But, issue of rights shares increases the outstanding shares in the market and this in turn dilutes value of the shares you hold. Please check out this post: What effect does rights issue have on stock price? - #3 by ShubhS9
So instead of not applying and letting the REs lapse, you can sell them in the market and pocket the premium.