Text::CSV (or another CSV parser) is your best bet for a reliable, robust solution. The issue primarily revolves about that fact that you need to count the number of quotation marks from the start of the string. For example, if your input were (check out Quote and Quote like Operators):

$inputstring=qq{field1\tfield2\t"field3\t\t"\tfield4\t"field5\t\t"\n};

you would not want to strip the tabs surrounding field4 even though it is surrounded by quotation marks. A regular expression like s/^([^"]*"[^\t"]*)\t+/$1/ would strip the tabs from the first quoted field, but not from all fields. You could wrap that in a while loop and swap the first character class to allow for paired quotes

while (s/^((?:[^"]|"[^"]*")*"[^\t"]*)\t+/$1/){1}

but CSV also includes an escape character, and any escaped quotation marks would then break the above. So why fight with all that when there is a well-tested solution at your fingertips?