If I sell a naked put on VLP, do I have to pay the partnership distribution if I'm holding the option when the distribution is declared?
No you don't. It's often easier to take an example. Imagine you sell a put option, strike $100 on a MLP trading at $100, expiring in a month and the market expects a dividend payment of $5, ex in 5 days. Then the option will price based on the fact that the stock will probably trade around $95 when it expires and your put will be worth $5 + some time value.
On the ex date, the stock will go down by ca. $5 but the price of the option won't move due to that price move because it was already "priced in".
If however the company announces the day after you bought/sold the option that the dividend will be $6 instead of the expected $5, your put will probably raise by $1 to reflect the new expectation.
Whether you buy or sell the put, you won't pay/receive any dividends.
No, if you write (or sell) a naked put option, you receive the premium, pay brokerage and an initial margin. If the underlying's price moves in your favour (up in this case) your margin requirements will be reduced, if the price moves against you (down) your margin requirements will increase. The only other thing you may have to do is buy the underlying at the strike price off the put option buyer if he/she exercises their options.
You do not have to pay any distributions or dividends when you write an option, however, distribution or dividends may affect the underlying price and in effect the price of the options.