What is the ruling on lying to a sick mother suffering from heart disease, fearing her condition would worsen?

It is not permissible to lie in this situation. Instead, one can resort to Tawriya (ambiguous speech) and Ma'areedh (allusive speech). For example, you could say, "So-and-so is fine," intending that he was fine yesterday. In such expressions, there is an escape from lying. If you need to swear an oath regarding the true meanings you convey, there is no harm in it, provided that no right of another person is involved, such that your ambiguous speech and oath prevent that right from being fulfilled.