First off just a little preface: Most of of my advice is based on my subjective experiences with Paypal. I may be entirely incorrect on some of the assumptions I make, so take the following with a few pinches of salt.
So dealing with payments to help with the running costs of a Minecraft server is something that a sizable amount of the community will have come up against at some time or another. Chances are you've read before about server owners being hit with chargebacks or paypal holding onto large sums of money for extended periods of time.
This post is about things you should and should not do as a server owner to help avoid some of the worst problems, and some advice about dealing with common issues you'll come across.
Dealing with payment disputes and chargebacks.
One of the common issues the Minecraft community has to deal with is younger players using their parents credit cards to buy benefits on servers without their permission, and then the cardholder invoking a chargeback with their credit card provider, or a dispute on paypal for unauthorized use.
Chargebacks are the bane of everyone who uses Paypal to accept payments. They're extremely difficult to fight, and worse they come with a hefty £14 GBP/~$20 USD fee if you don't win them, even though you're not at fault.
If the cardholder raises the dispute with Paypal you're very lucky. You'll lose the money originally given to you, but that's it. Paypal even refund the fees. For virtual goods there's no seller protection so there is still the problem where someone can donate, then raise a dispute saying it was unauthorized, and get their money back. This doesn't happen often though because paypal will not allow an unauthorized transaction dispute in the particular case of a family member using your card. Hence parents are very likely to just jump straight to the credit card company who will.
This is where things start to really suck. Since you're essentially selling virtual goods, it's almost impossible to prove receipt/delivery to the degree that will satisfy the credit card company. Any which point you're hit by the nasty fees mentioned earlier.
Now there are some ways to help mitigate this problem:
For larger donations (Over about £20/$30), consider sending a 'thankyou' card, voucher, or something physical to the billing address via recorded delivery. This allows you to prove where the transaction came from, which makes dealing with disputes much much easier. This of course doesn't work if you're handling higher volumes of smaller amounts.
Discourage minors from donating. This is a bit of an issue since they are likely to make up a large amount of your player base. Consider requiring users to register before they can donate and require their age. If they're under 18, require that they provide a signed parental consent form. You can tie their registration to their Minecraft username to prevent someone registering twice once they find out that there is an age restriction.
Now this will cost you perfectly legitimate donations that you might have gotten otherwise due to the extra hassle. So you'll need to carefully consider the balance between income and risk to determine what barriers you're willing to put in peoples way.
Don't use the 'D' Word
This is a word you should avoid at all costs, not because it's really that inaccurate - but because it's a very easy way to have your funds frozen.
That word is 'Donation' or 'Donate'.
Paypal and other payment providers get very uptight if you use those words but aren't actually a registered charity or non-profit (Google checkout won't even let you take the payments unless you provide documentation proving you're a non-profit). Yes, the reality is that players are in effect donating towards the upkeep of the server with nothing real in return for it - but avoiding those words will save you a lot of pain. There's also legislation in certain countries that dictates donations have to be refundable within a certain amount of time or if they're over a certain amount - so keeping distanced from that will save you a lot of hurt.
So instead you can call it a "Contribution" or invite users to "Contribute" and give them a rank of "Contributor", and on any websites use the "Pay now" button instead of the "Donate" button to the effect that users are buying access to the "Contributor" rank on your server, along with any benefits you decide to bestow on them for this.
Do run your server like a business
This may seem counter intuitive for something that's just meant to be for fun, but it goes a long way to keeping everything manageable. Incoming contributions are your revenue source, the server is an expense, and dealing with payment issues is a cost of business that you need to allow for (see the section earlier on chargebacks).
In this vein you need to keep an eye out on cash flow. If 100% of your incoming revenue is used up the moment your server bill comes in, then you've got a cash flow problem because all it would take is for a single payment to be reversed (or worse, to have a charge back issued on the credit card) then you're immediately out of pocket.
So make sure you've always got a persistent balance available at all times, at least 10% of your monthly revenue should be put to one side each month to allow for either sudden costs that could jeopardize the server or just month-to-month inconsistencies. Just like real life you don't want to be living paycheque-to-paycheque. If you can't afford to do this then you should start either finding new revenue sources, or consider downsizing your server if you're not able/willing to make up the difference out of your own pocket.
Also like a business, look after your 'customers'. Consider rewarding those who donate regularly by sending them real-life gifts (or even just a thank you card) or other benefits to make them feel good about donating and keep them donating. Good will is an asset.
In addition, making your expenses public to your users will help build trust that their money isn't being squandered. Let them see how much your server bill is, and any other costs associated. Transparency is key to building a good relationship with players who are giving you money to run your server.
Don't let users buy their way into power
While letting users buy ranks that give them influence over other players is a near sure-fire way to get extra donations, it's also the best way to stop anyone else from donating and increase the number of chargebacks and disputes you get. This may seem really obvious but it's still something that far too many servers try and do.
The reasoning for this is pretty straightforward. It makes it too easy for one user to put themselves in a position where they can abuse other players, which will very quickly reduce your servers population and the number of people willing to contribute financially. Then when you find out that someone has abused their power and you revoke it - you can be reasonably sure that they will file a dispute out of spite.
This shouldn't even need to be a point here. It's suicide for your server. Don't do it.
Do keep an eye on the tax situation
This varies a lot from country to country (or even state to state in the US) so I'm not going to talk about this too much. The main thing is that you consult a tax adviser and make yourself familiar with your local tax laws to check what you need to do.
In the UK for example, if you're running the server like a business then you may need to declare yourself as self-employed (in addition to any normal day job you have) and you will likely have to pay taxes on your profits (income minus costs) - you still need to do this even if you don't make any profit. This also means filling in a self-assessment tax at the end of the year to declare your earnings and cost from the business.