Here are some facts that an attorney might want to ask/know before advising a tenant in the situation described in the question.
1. Do you have a lease or a month to month? Absent having a lease, or perhaps having moved in with a month to month after the default occurred, it's hard to see how the tenant would have any damages from the foreclosure of the landlord's interest. They would had no contractual right to stay beyond the notice provisions required by law, and thus no damages. This might be different in Seattle, where the landlord is more limited as to the right to kick a tenant out.
2. If it is a lease, when does it expire? If it expires prior to the foreclosure date, and there's no option to renew, then again it's hard to see how the tenant was damaged. But again, if it was entered into after the default, perhaps there are other claims the tenant could make. Move in/move out costs might become an element of damages.
3. How does the lease rate compare to market rates, and what are the costs associated with moving? Those are questions that would go to whether the tenant has any significant damages from the potential foreclosure. If say the lease was a sweetheart deal, then they might have significant damages. If the lease rate was extremely high, or perhaps a lease entered into by a couple where one has since left, the lease terminating early might actually be a good thing for the tenant.
4, What is the date each monthly term starts, and how does that compare to the foreclosure date? This relates to the last month where the tenant does have some real concerns. I believe Washington law allows 20 days for the tenant to stay after the sale. Assuming the monthly period starts the day before the sale date, then the tenant could risk losing about 10 days rent if the property does end up getting foreclosed. Some decisions might have to be made there. For example, perhaps paying a late fee to make sure the sale does not occur on the scheduled date before paying rent. But if that is done, then a prospective landlord might be turned off by even the one period of paying rent late. There's some risk to that course of action.
5. Has the tenant talked to the landlord, and if so, what have they said? About the only obvious claims of the tenant that come to mind are that the tenant might have would be either a fraud claim (where the tenancy was started after the default) or an anticipatory breach of contract claim. Both issues are problematic, and so it would be best to know the landlord's stated plans and explanations.
6. Does the lender have an assignment of rents clause, and if so have they done anything to collect on it? To answer the first part the attorney would probably need to contact a title company, and then research the current status of the law assuming no action has been taken by the lender.
7. How large is the building being foreclosed? If a single family residence it's much more likely the new owner would require the tenant to move than if the building is a 40 unit apartment house. Empty apartment houses are not very valuable.
The bottom line is that the question asked does not contain sufficient detail to know the answer, but it is unlikely that absent something more than a simple notice of trustees sale that an attorney would advise a client to simply quit paying rent because the landlord was in default. That default alone probably does not entitle the tenant to a 3+ month period of not paying rent. Also, assuming the attorney does think that the tenant does have some sort of a claim, they'll need to assess the amount of that claim and its strength. Absent paying rent the landlord may very likely start an unlawful detainer case against the tenant, which could be expensive to defend and affect their ability to get new housing in the future. The course of least resistance would be to simply pay the rent and then move when the time comes.
Thus, the question asked does not have a simple answer, and anyone facing that situation would need to consult their own attorney (or perhaps a tenant services group). The answer the attorney gives could turn on any one of the questions above, or another question. Or it might turn on a city ordinance. Each case would be fairly fact specific.