The word is independent of tense.

You don't need to look at the word "however". Pay attention to the tenses in the rest of the sentence. If you want to put a verb after "tough" in the first sentence, then "be" is your answer. Basically, you've removed the modal (eg. Tati's suggestion of "may") and you have a simple present subjunctive. Obviously, "was/were" can't match with "will be", and "is" is the indicative form, ie. a real action, as opposed to the imagined actions in your sentence. For the second sentence, either "is" or "be" will fit. If the truth really is harsh and you know it, use "is". If the harshness is unknown and you're speculating, use "be". Note that using "be" here sounds a little formal in tone. That's subjunctive for you.

I don't know why but I feel like putting MAY here will do nicely. - You will be able to cope with any problem, however tough it may be. - You'd better tell the truth, however harsh it may be.