Politics: Emoluments Clause
Here are a few items you could check out about this matter:
Here is an item from ConstitutionCenter.org giving the history of how the clause has been complied with over the years
WaPo legal opinion/analysis item from Volokh Conspiracy legal affairs blog
WaPo item from their news section giving a rundown of the lawsuit that has been filed
Am not a Constitutional scholar nor do I play one in the movies but my gut reaction is that the lawsuit will probably fail for the reasons mentioned in some of the above articles:
Standing issues - the plaintiff has to be harmed in some way to have standing for a lawsuit.
Non-applicability - routine business activity may not be covered under the clause.
Ultimately, the goal for Trump critics is to use the clause as a way to initiate impeachment and/or force a resignation. However, impeachment is not lightly undertaken. In the history of the US, only two Presidents (Andrew Johnson and Bill Clinton) were impeached by the House and both acquitted in subsequent Senate trial. Richard Nixon would have been impeached and convicted but he resigned.
Down the road, I suppose if it can be showed that foreign governments vastly overpay for the services of the Trump hotels, then the claim could be made that those overpayments constitute attempts to influence. At that point, it would be up to the House to decide if they view those attempts to influence to be so serious as to initiate impeachment proceedings. It would have to be shown that the financial benefits had the effect of causing treasonous activity by the President that is injurious to US national interests or were a form of bribery such that the giver receives benefit from the President.