Usually. However, we do handle these issues on a case-by-case basis, considering a number of important factors.
First, we look at who is the target of the court order. Some court orders are directed at you as the site owner, and some are directed at your service provider.
If the court orders us to take the material down, and that court has jurisdiction over us, the material comes down. We uphold United States law at all times, end of story. If the court does not have jurisdiction over us, we will typically request that the order be domesticated before implementing it.
If the court orders you to take the material down, it's a bit more complicated. We will then consider whether the court has jurisdiction over you, either due to citizenship or residency. If they do, then we will typically prohibit you from using our service to violate the court order. If there's a court order against your content, you need to fight the order, not try to use the Internet to evade it.
Some countries' legal systems now include the theory that their courts can exercise jurisdiction over any content on the Internet that is visible from inside that country. We repudiate that theory, and do not accept orders from courts that cannot exercise personal jurisdiction over us or you unless they have been properly domesticated, and domestication of such orders is frequently disallowed by US law. (However, if Elbonia instead requires all Elbonian ISPs to block access to your site, that is not something we're likely to be able to help with.)
We will also consider the free speech implications of the order. However, many people don't understand how limited this is. Typically, we will only consider free speech based arguments when the subject of the material is the government issuing the court order, or if the content is primarily political in nature. I.e. we would be very likely to disregard an Elbonian court order requiring removal of content critical of the government of Elbonia. We would be significantly less likely to disregard an Elbonian court order requiring you to remove material that the Elbonian court found to be defamatory of an Elbonian citizen if you are also an Elbonian citizen or resident. Using our service is not a way to opt out of your country's laws.
While we will give some consideration as to whether a foreign court order is a travesty of justice, it is definitely one of the least influential factors and, as a practical matter, it rarely comes into play.
One important factor we do not consider at all is your opinion. This includes your opinion of how justified your actions are, how messed up your country's laws are, how easy/unfair/one-sided you feel it is to get such an order in your country, what an ignorant jerk you think the judge was, or how sure you are that they're all out to get you. If you agreed with the order and the process, you'd take the material down yourself and we'd never hear about it. Beyond that, your opinion ceases to have any probative value. Sorry if that's hard to hear.
When we deal with court orders from your jurisdiction, and you are outside the US, it is often helpful for you to make your attorney available to us (at your expense) to answer questions about the matter. If you got a court order issued against you in your own country, but you don't have qualified legal representation, it will be somewhat harder for us to take you seriously.
To reiterate the most important statement at the top, all situations are handled on a case-by-case basis. We will definitely not guarantee any specific response, nor will we even guarantee that all cases will be handled as described here. However, we have been at this for many years, and the guidelines above have well served our goal of staunchly defending freedom of expression without letting the Internet collapse into total anarchy.
In all cases, the indemnification provisions of our Terms and Conditions of Service require you to pay any legal expenses we incur in handling or responding to court orders related to your services with us.