The Dutchman, who replaced Allardyce at Selhurst Park in the summer, was sacked after just 77 days in the job after Palace claimed the unwanted title of becoming the first top flight side in 93 years to lose their first four games without scoring.
"I have just heard the news that Frank has left today which is obviously very disappointing for him and for Palace," Allardyce told Sky Sports News.
"But it shows the panic in football today is ever increasing due to the size of the money and the fear of losing your position in the Premier League.
"As a manager, you have to live or die by your results and unfortunately now that is quicker than ever. It is a great shame that Palace have not continued where I left off."
Palace delivered a much improved performance against Burnley on Sunday (10 September), but their profligacy in front of goal cost them dear again as they slumped to a 1-0 defeat at Turf Moor. And Allardyce, who steered Palace to safety last season before retiring from management at the end of the campaign, said the Eagles could count themselves unlucky after failing to pick up a point.
"I watched them yesterday and I have to say the fact they actually lost the game was beyond me," he added.
"Twenty-two attempts at goal, complete domination but unfortunately the guys at the end, when the chances came, couldn't put the ball in the net.
"Four games with no points makes life extremely difficult for Palace."
The Eagles' chairman, Steve Parish, was understood to be reluctant to part ways with De Boer only four games into a new season but the latter's tactical rigidity won him few friends in south London.
The Dutchman's preferred 3-4-3 line-up and possession-based football attracted criticism, particularly as the group of players he has inherited from his predecessor appear to be at odds with the system.
However, Parish has since attracted criticism for not giving De Boer more time but Allardyce, who last month ruled himself out of a return to the Premier League following interest from Palace, had sympathy for his former employer.
The last thing Steve, Josh [Harris] and Dave [Blitzer] would want is to be making a decision like this so early in the season," he said.
"But obviously they feel for whatever reason it is the right thing to do.
"Every manager that is not picking up points from here on in will suffer the same pressure until they turn the results around."
Roy Hodgson has since emerged as the favourites to replace De Boer, with a number of outlets indicating the former England manager will sign a two-year deal at Palace. The 70-year-old, who left his post as England manager after a dismal showing at the Euro 2016 which saw the Three Lions knocked out by Iceland in the round of 16, has not managed in the Premier League since guiding West Bromwich Albion to a 10th place finish in the 2011-12 season.