which is better, "We hope our close partnership can continue well into the future" or "We hope our close partnership can be continued well into the future"?
As @Lamar Latrell comments, it may be best to use will continue.
Of the two options given in the question, though:
"We hope our close partnership can continue well into the future"
is better. I say better, although both options make good sense, and are fine grammatically. It's just a matter of the nuanced implications of the sentences.
Using the passive voice, "can be continued", suggests that there is some doubt about the possibility of its continuing, and that there is effort involved in making it continue (that one or more of the parties in the partnership is the implied agent who makes it continue).
"can continue" is more suggestive that the partnership will continue naturally, as a matter of course (although there is still a chance that it won't).
So the most friendly, positive statement is to hope that it will continue.
Conversely, if you do in fact want to express doubt about the partnership, like if the person you're writing to has been breaking the terms of a deal, it may still be better to use the active voice and make your concern explicit: the passive form can sound passive-aggressive. Something like:
We hope you will work more closely with us in future, to continue our partnership...