The rights issue is an offer that needs to be accepted by the shareholder. In case, a shareholder does not subscribe to his rights, the rights get lapsed i.e. the shareholder will not get any additional shares if not applied for.
The shareholder's ownership in the company will get diluted as their stake would get reduced by not applying to the rights issue. Moreover, the ex-rights issue price generally comes down (price fall expected due to an increase in the number of shares, however, it all depends on the market sentiments and price may not fall). Thus if a shareholder does not subscribe to the issue he may tend to lose the compensation offered by the discounted price. Hence, one must keep a track of the company's performance, prospects, and the reasons for raising the capital and then decide whether or not to subscribe to the rights issue.