Once you exercise an esop, it converts to a share. Even if you quit the company, you remain its shareholder.
To exercise an esop, you need to pay the exercise price (which is generally 1 rs per share) . The actual current value of esop then converts to a perk for you and is taxed according to your Income tax slab so your company will ask you to deposit tax amount so they can deposit it as tds.
If the current value is low and you think the value will grow in future and you will get good returns, you can do the exercising - the cost for you is income tax that you will have to pay. It’s great if company is doing an immediate buy back or going for ipo soon. One benefit of exercising early in this case is that if price of shares increased significantly you will have to pay lower capital gains tax (20% for short term capital gain and 10% for long term. For private companies holding a share for 2 years make it eligible for Long term and for public listed, duration is 1 year).
On the other hand, if the future is bleak, it’s possible that share price at which you are able to finally sell the exercised shares is lower than current value (or zero, if company dies) then your loss is the income tax that you would pay.