It varied enormously. It doesn't seem to have impacted Da Vinci at all (I actually wrote about his early life and family situation, among other things, here) while other children born out of wedlock might be entrusted to an orphanage, like the Spedale degli Innocenti in Florence (which still provides childcare to this day) or other religious institutions.

Most major cities had at least one institution called "Brefotrofio" taking on the task of raising unwanted children, many of whom were born out of wedlock. In Milan, unwanted infants born out of wedlock were entrusted to the friars of the Church of San Salvatore in Xenodochio (adopting a word commonly used to denote accommodations for religious pilgrims). In Rome, they were entrusted to the Church of Santo Spirito in Sassia. If, instead of entrusting the child to a Brefotrofio, a family chose to recognize a child born out wedlock, I very well suppose they might attract resentment and/or embarrassment. However, that's far from a foregone conclusion: just look at the aforementioned Da Vinci. Although I will also say that Da Vinci's father was well-off and squarely upper-middle-class. That would certainly have helped mitigate any gossip. But again, the point is that there's no definitive rule or trend: there are just too many variables at play in each individual situation. Sure, many times children were entrusted to a Brefotrofio because the mother didn't have the means to raise the child herself, and she might have been shamed out of trying to do so. However exceptions aren't rare.