Is our lying to our mentally ill father to avoid the deterioration of his health considered a sin?

Lying is fundamentally forbidden, unless there is a legitimate benefit, or a warding off of harm, or reconciliation between disputants. In such cases, it becomes permissible, because speech is a means to achieve objectives. Any commendable objective that can be attained without lying makes lying forbidden in that instance. However, if it cannot be achieved except through lying, then lying becomes permissible. Thus, if the attainment of the objective is permissible, then lying is permissible; and if it is obligatory, then lying is obligatory. In your father's case, if his going out does not result in harm, then take him out wherever he wishes, as long as it does not harm you or him. But if his going out would result in harm, then lying to him is permissible to avert these harms. The safer approach, however, is to use tawriyah (ambiguous speech).