The first step is to ask Perl to tell you about errors:

    open(INFILE, "PSR_Date.txt")
        or die "Couldn't open input file 'PSR_Date.txt': $!";
    open(OUTFILE, ">PSR_OutputDate.txt")
        or die "Couldn't create output file 'PSR_OutputDate.txt': $!";
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I did do this in regards to verifying the files. I removed the code and everything worked fine. I ended up finding my mistake, I was missing "my" in front of "$line" Thank you for the help.

See also "open" Best Practices.

ALWAYS CHECK THE RETURN FROM open.

my $infile = qq{PSR_Date.txt};
my $outfile = qq{PSR_OutputDate.txt};

open( my $infh, q{<}, $infile ) or die qq{Can't open '$infile' to read
+: $!\n};
open( my $outfh, q{>}, $outfile ) or die qq{Can't open '$outfile' to w
+rite: $!\n};

while( defined( my $line = <$infh> ) ) {
  ## ...
  print $outfh $line;
}
close( $infh );
## Note that depending on OS, amount written, buffering, phase of moon
+ you might not see output in the file until you do this.
close( $outfh );
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Also, it looks like you are looking for dates in YYYY-MM-DD format, yet your data is in MM/DD/YYYY format. You're also going to get just the first date in each line, but I'm guessing that was your intent.

NB: The  \1 \2 \3 notation is for regex back-references and is not kosher in a  s/// replacement string. warnings would have told you this had you enabled them. This (ab)use of back-references is a special dispensation of Perl, but should be avoided in favor of capture variables; see Variables related to regular expressions in perlvar.

c:\@Work\Perl\monks>perl -wMstrict -le
"use strict;
 use warnings;
 ;;
 my $line = 'OD44202,07/01/2015,08/19/2019,08/27/2019,1,07/01/2015,06/
+30/2019,2015-06-20 00:00:00';
 ;;
 $line =~ s/(\d{4})-(\d{2})-(\d{2})\s\d{2}:\d{2}:\d{2}/\2\/\3\/\1/g;
 print qq{>$line<};
"
\2 better written as $2 at -e line 1.
\3 better written as $3 at -e line 1.
\1 better written as $1 at -e line 1.
>OD44202,07/01/2015,08/19/2019,08/27/2019,1,07/01/2015,06/30/2019,06/2
+0/2015<
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c:\@Work\Perl\monks>perl -wMstrict -le
"use strict;
 use warnings;
 ;;
 my $line = 'OD44202,07/01/2015,08/19/2019,08/27/2019,1,07/01/2015,06/
+30/2019,2015-06-20 00:00:00';
 ;;
 $line =~ s{(\d{4})-(\d{2})-(\d{2})\s\d{2}:\d{2}:\d{2}}{$2/$3/$1}g;
 print qq{>$line<};
"
>OD44202,07/01/2015,08/19/2019,08/27/2019,1,07/01/2015,06/30/2019,06/2
+0/2015<
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Update: Your eyes will also thank you for some whitespace courtesy of the  /x modifier:
    $line =~ s{ (\d{4}) - (\d{2}) - (\d{2}) \s \d{2} : \d{2} : \d{2} }{$2/$3/$1}xg;
And is the  /g modifier really necessary?

The code as presented can not have run so the issue you describe is not with that code. Please make sure that the code that you post is the code that you ran. Very often "tidying" the code for publication will turn up the issue if you run the code.