How to pass variable in pattern while using sed command?

As long as the numbers in your example does not contain the delimiter that sed is using (by default /), the $p in your code will be interpreted as a regular expression (with all what that means).

Your code:

search_dir='dummy'
filename='numbers.txt'

for entry in "$search_dir"/*
do
  while read p;
  do 
    sed -i '' "/$p/d" $entry
  done < $filename
done

Here, you want to delete all lines in the files under $search_dir that contains any of the numbers in $filename. Whether this work or not depends on how your sed treats -i ''. With some implementations of sed you would have to use -i without an argument.

Related to sed -i and portability: How can I achieve portability with sed -i (in-place editing)?

It is safer to write the result to a temporary file and then to move that file to the original filename:

for entry in "$search_dir"/*
do
  while read p;
  do 
    sed "/$p/d" "$entry" >"$entry.tmp" && mv "$entry.tmp" "$entry"
  done <"$filename"
done

This ensures that it will work regardless of what sed implementation you happen to be working with. In general, it's a bad idea to try to make in-place changes to files while testing out a script, so you may well want to comment out that mv before you are happy with the way the script otherwise works.

This is still a bit unsafe as a general solution though since you're actually "using data as code" (the numbers are data, and you use them a part of your sed script). This means that you easily could cause a syntax error in the sed script by just inserting a / in one of the numbers in your numbers file.

Since the operation is so simple, we may instead use grep. This also gets rid of the inner while loop:

for entry in "$search_dir"/*
do
  grep -Fv -f "$filename" "$entry" >"$entry.tmp" && mv "$entry.tmp" "$entry"
done

This will cause grep to read its patterns from $filename and to apply these to the $entry file. The -v means we'll discard any line containing the pattern and -F means grep will not interpret the numbers as regular expressions but as fixed strings. With -f "$filename" we get grep to read the strings from $filename.

If there may be directories under $search_dir we would want to skip these:

for entry in "$search_dir"/*
do
  [ ! -f "$entry" ] && continue
  grep -Fv -f "$filename" "$entry" >"$entry.tmp" && mv "$entry.tmp" "$entry"
done

Another, even safer way to do your operation is to use awk. Since with both the sed and grep solutions above, the number is matched anywhere on the line, it is conceivable that we might delete the wrong lines. With awk it's easy to match just the second ~-delimited field in the data:

for entry in "$search_dir"/*; do
    [ ! -f "$entry" ] && continue
    awk -F '~' 'NR==FNR { num[$0]; next } !($2 in num)' "$filename" "$entry" >"$entry.tmp" &&
    mv "$entry.tmp" "$entry"
done

The awk program first populates an associative array/hash with the numbers as keys, and then prints every line from the $entry file whose second ~-delimited column is not a key in that hash.