No Grand Slam tournament is exempt from the wrath of nature, including the French Open. The claycourt Major has dealt with its fair share of weather delays, mainly due to heavy rains. Even the ongoing edition of the tournament has been hit by rains, but the tournament schedule hasn't gone for a toss just yet.
The French Open organizers came under heavy fire in the past for not having effective countermeasures in place to deal with torrential rains. It was the last of the four Majors to have a retractable roof on its premises, with Court Philippe Chatrier, the main show court, getting one in 2020.
Court Suzanne Lenglen, the second biggest court at the venue, also has a retractable roof now, the construction of which was recently completed. The call to close the roof is taken by the chair umpire, and play is suspended until the roof is completely closed.
However, matches on outside courts are still vulnerable to adverse weather. The courts are covered once it starts raining, and play is suspended until the weather clears up. The courts are then dried before players are called to continue the match.
However, if there are no signs that the heavy downpour is going to subside, all matches are called off for the day. With the addition of roofs to the venue's biggest courts, a match from an outside court can be moved to a court with a roof.
With the addition of a roof to Courts Philippe Chatrier and Suzanne Lenglen, the French Open is expected to conclude on schedule

While matches can get pushed to the next day due to bad weather, that's mostly a concern during the first week of the French Open. With a ton of matches to wrap up in the initial days, the schedule can go sideways until the second week if it rains too heavily.
However, as the field gets thinner with every round, the question of postponing matches to the next day doesn't arrive towards the end of the tournament. With two of the tournament's biggest courts now having a retractable roof, matches can continue regardless of the weather outside.
That's especially true for the championship weekend, as all matches take place on Court Philippe Chatrier, which has a roof. The addition of a roof to Court Suzanne Lenglen means that the junior championships during the same time can also continue uninterrupted by the weather conditions.