After the Catalans beat Girona on Saturday Madrid needed to win to stay three points behind their rivals in the title race, but Remiro ensured otherwise, keeping third-place La Real on the champions' tail.
Vinicius Junior had three good chances repelled by Remiro among other Madrid openings, as Carlo Ancelotti's side sparkled but failed to find the goal their performance deserved.
The draw continues a mixed start to 2023 for Real Madrid, slipping further behind Barcelona in the league and losing against them in the Spanish Super Cup final, but reaching the Copa del Rey semis on Thursday with an extra-time win over Atletico Madrid.
"It was one of the best games we've played this season," Ancelotti told reporters.
"It's true Barca are doing very well, but the season is very long, January is very complicated, but the team have progressed a lot in all areas, in defence, attack, physically.
"The players are starting to improve, the injured players (will return), so we are confident for the coming months, we can get stuck into it in the second half of the season."
Madrid constructed some slick moves in the first half, with Brazilian wingers Vinicius and Rodrygo incessant threats.
The former curled inches wide after a sensational nutmeg on Igor Zubeldia, while the latter teed up Karim Benzema with a backheel, but Remiro saved comfortably.
Ancelotti started with Eduardo Camavinga playing out of position at left-back because of injuries elsewhere, and the former Rennes midfielder thrived.
Vinicius should have opened the scoring before the break but his shot was well kept out by Remiro, the visitors' last line of defence.
Remiro made another smart save from the Brazilian winger early in the second half with his leg to keep his team level, while Fede Valverde hammered a shot wide from distance as Madrid flowed forward.
Real Madrid goalkeeper Thibaut Courtois was called into serious action for the first time after an hour when Takefusa Kubo broke free on the right, but the Belgian got down well to deny him.
- Unbeatable Remiro -
Vinicius was foiled once more by the excellent Remiro, who pushed away the Brazilian's attempted lob after cleverly accelerating into the box.
Remiro was booked for time-wasting in stoppage time as he eked out every second at a goal kick to clinch a hard-fought point for Real Sociedad, still three points behind Madrid.
Earlier Saul Niguez's cool finish made the difference for Atletico Madrid as they held on to fourth place with a narrow 1-0 win at Osasuna.
Out of the Champions League and Copa del Rey, and 13 points behind La Liga leaders Barcelona, Champions League qualification is Atletico's key goal in the second half of the season.
The Rojiblancos were left fuming after their quarter-final cup elimination in midweek, unhappy with the refereeing, but had nothing to complain about as they snatched victory at El Sadar.
Atletico took the lead after 74 minutes against the run of play, through midfielder Saul, on as a substitute.
The Spaniard is no longer a key figure for Diego Simeone but took his chance, reaching Rodrigo De Paul's lobbed pass and finishing clinically.
Simeone believes his team bounced back well from the team's Copa del Rey elimination by Madrid.
"We have a way of thinking and working, and the club has a way of living that is to never give up, and always get up in the face of adverse situations," the Argentine coach told Movistar.
"There is no greater motivation than playing for Atletico Madrid."
Iago Aspas's goal helped Celta Vigo earn a 1-0 win over Athletic Bilbao, while a 90th-minute strike by Canadian debutant Cyle Larin helped Real Valladolid beat Valencia 1-0, moving a point clear of the drop zone.