my $data =">DL;H1_ENSP00000194530_chr2_202024 CCCC---GCCTTCTCGCTGCCCAG
+C--CCCGGGGAGGGAGG*";

$data =~ s/[-*]|DL;//g;  # globaly, replace all "-" or "*" OR "DL;" wi
+th nothing
                         # ignores possibility "DL;" appears elsewhere
+ in data
                         # if that's an issue, you might want to do tw
+o substitutions
                         # $data =~ s/[-*]//g;  and $data =~ s/(>)DL;/
+$1/g;
                         # though that last is NOT tied to the beginni
+ng of the line
                         # which appears to be subject to brain_block 
+at the moment
print $data;

=head OUTPUT

perl dataclean.pl
>H1_ENSP00000194530_chr2_202024 CCCCGCCTTCTCGCTGCCCAGCCCCGGGGAGGGAGG

=cut
[download]

Update: Credited the wrong person.

I'd suggest a slight modification to ww's method. As this is a FASTA file, I'd read the file record by record (sequence by sequence), rather than line by line. The following one-liner ought to work, but is untested.

perl -e"BEGIN{$/=qq[\n>]}" -wpe"s[[-*]|DL;][]g" theFile > theOuput
[download]

Tested (albeit, non-rigorously) BrowserUK's with a data file (very modestly varied from OP) eq:

>DL;H1_ENSP00000194530_chr2_202024 CCCC---GCCTTCTCGCTGCCCAGC--CCCGGGGA
+GGGAGG*
">DL;H2_ENSP00000194530_chr2_202024 CCCC---GCCTTCTCGCTGCCCAGC--CCCGGGG
+AGGGAGG*" line 2
>DL;H1_ENSP00000194530_chr2_202024 CCCC---GCCTTCDLTCGCTGCCCAGC--CCCGGG
+GAGGGAGG* line3
>DL;H1_ENSP00000194530_chr2_202024 CCCC---GCCTTCTCGCTGCCCAGC--CCCGGGGA
+GGGAGG line 4
>DL;H1_ENSP00000194530_chr2_202024 CCCC---GCCTTCTCGCTGCCCAGC--CCCGG*GG
+AGGGAGG line 5
[download]

and output is:

>H1_ENSP00000194530_chr2_202024 CCCCGCCTTCTCGCTGCCCAGCCCCGGGGAGGGAGG
">H2_ENSP00000194530_chr2_202024 CCCCGCCTTCTCGCTGCCCAGCCCCGGGGAGGGAGG"
+ line 2
>H1_ENSP00000194530_chr2_202024 CCCCGCCTTCDLTCGCTGCCCAGCCCCGGGGAGGGAGG
+ line3
>H1_ENSP00000194530_chr2_202024 CCCCGCCTTCTCGCTGCCCAGCCCCGGGGAGGGAGG l
+ine 4
>H1_ENSP00000194530_chr2_202024 CCCCGCCTTCTCGCTGCCCAGCCCCGGGGAGGGAGG l
+ine 5
[download]

Nice, BrowserUK; ++

Update: Fixed the mis-attribution. Give BrowserUK another ++ and I'll do penance in the dungeon; the more so, since it was he who answered a brain_dead question about his code.

What bit are you having trouble with? What have you tried already? Can we see some (short) code?

I believe you want that:

DL;H1_ENSP00000194530_chr2_202024 CCCC---GCCTTCTCGCTGCCCAGC--CCCGGGGAGGGAGG*

To became:

H1_ENSP00000194530_chr2_202024 CCCCGCCTTCTCGCTGCCCAGCCCCGGGGAGGGAGG

A very naive approach would be using tr and s to remove the undesired characters, like this:

# in a loop while reading
$_ =~ tr/-//d;
$_ =~ s/^DL\;//o;
$_ =~ s/\*$//o;
[download]

Of course, I'm unaware about any rule to remove those characters, since you didn't mentioned anything about it.