Suppose a student currently doing Ph.D in an institute wants to leave the institute and apply abroad for Ph.D. Should he/she mention the name of the institute he/she wants to leave? If the universities call the current PI, they will never get a positive feedback since the current institute will never want to lose their student. But the student wants to mention that he/she has work experience. What is to be done in this situation? Can the US universities reject the application based on the current PI's feedback in spite of the candidate having excellent grades and recommendations from previous PIs?
Should he/she mention the name of the institute he/she wants to leave?
Yes, absolutely. The application will typically ask for your complete academic history, which would include the institution where you are currently studying. Leaving it out would be an extremely serious ethical violation that could jeopardize not only your chances of admission but your entire academic career.
If the universities call the current PI
They normally wouldn't call your current PI. But one of the requirements to apply to a US graduate program is to have letters of recommendation from people who have known you academically. If you are currently in a PhD program, they would usually expect that one of those letters would be from your current supervisor, since that is the person who would be most familiar with your most recent work. If you don't have a letter from them, an admissions committee might make the assumption that your current work is not very good, and so you didn't request a letter from your supervisor because you knew it would be unfavorable.
You can mitigate this somewhat if you have a letter from someone else at your current institution, especially if their letter explains why you could not ask the PI. But your application probably still won't be as strong as if you had a positive letter from the PI. If you have no letters from anyone at your current institution, that's likely to be a red flag.
Many PIs would have no problem writing a positive letter for a student wanting to make a move, and I think any PI who refuses to do because they don't want to lose the student is being selfish and doing harm to their profession. But you know your PI better than we do. So you're going to have to make your own judgment as to the relative risks and benefits of asking your PI for a letter, or seeing if there is someone else there whom you trust.
Can the US universities reject the application based on the current PI's feedback in spite of the candidate having excellent grades and recommendations from previous PIs?
Yes, they can. The admission committee looks at the entire application, and any piece of it could affect their decision. At some very competitive departments, any significant piece of negative information on the application might be enough to reject it. Other departments might be willing to overlook a single negative (or missing) recommendation if other parts of the application are strong. There is no way to know in advance what any given committee would decide to do.