How do you say no to a funeral invitation without causing offense?

It depends on the circumstances. Good manners is all about being unselfish and considerate to others. That is particularly important when they are grieving.

If you cannot attend then you could send a letter of condolence, saying that you are very sorry for their loss, say something nice about the deceased, and offer the bereaved your best wishes.

As part of that include a line such as, "I am so sorry that I will not be able to attend Joe's funeral, but I will be thinking of you all on the day."

There is no need to explain why you cannot attend. You may unintentionally cause offence if your reason for not attending does not seem important to the bereaved. The most important thing is to show that you are thinking of them.

You could also either send flowers or a donation to the charity chosen by the family.