How do I remove the newline from the last line in a file in order to add text to that line?

If all you want to do is add text to the last line, it's very easy with sed. Replace $ (pattern matching at the end of the line) by the text you want to add, only on lines in the range $ (which means the last line).

sed '$ s/$/ and Linux/' <file >file.new &&
mv file.new file

which on Linux can be shortened to

sed -i '$ s/$/ and Linux/' file

If you want to remove the last byte in a file, Linux (more precisely GNU coreutils) offers the truncate command, which makes this very easy.

truncate -s -1 file

A POSIX way to do it is with dd. First determine the file length, then truncate it to one byte less.

length=$(wc -c <file)
dd if=/dev/null of=file obs="$((length-1))" seek=1

Note that both of these unconditionally truncate the last byte of the file. You may want to check that it's a newline first:

length=$(wc -c <file)
if [ "$length" -ne 0 ] && [ -z "$(tail -c -1 <file)" ]; then
  # The file ends with a newline or null
  dd if=/dev/null of=file obs="$((length-1))" seek=1
fi

Though, you can remove newline character from line by tr -d '\n':

$ echo -e "Hello"
Hello
$ echo -e "Hello" | tr -d '\n'
Hello$

You can remove the newline character at the end of file using following easy way:

1. head -c -1 file

   From man head:

   -c, --bytes=[-]K
             print the first K bytes of each file; with the leading '-',
             print all but the last K bytes of each file
   

2. truncate -s -1 file

   from man truncate:

   -s, --size=SIZE
             set or adjust the file size by SIZE
   

   SIZE is an integer and optional unit (example: 10M is 10*1024*1024).
   Units are K, M, G, T, P, E, Z, Y (powers of 1024) or KB, MB, ... (powers of 1000).
   
   SIZE  may  also be prefixed by one of the following modifying characters: 
   '+' extend by, '-' reduce by, '<' at most, '>' at least, '/' round down to multiple of, '%' round up to multiple of.
   

You can achieve this with perl as:

perl -pi -e 'chomp if eof' myfile

Compared to truncate or dd this not gonna leave you with a broken file if myfile had actually no trailing newlines.

(the answer is constructed from the comment, and based on this answer)