Fizzbuzz some Fibonacci numbers
My suggestions focus on style issues and pointless code and assume you're using a modern compiler.

In the identification division and environment division:


Delete the comment paragraphs. They were deleted from the standard 15 years ago and have been obsoleted by version control systems.
Rather than use the TITLE statement, name your program FIZZBUZZONACCI. All contemporary compilers support program-names longer than six characters.
Delete the non-functional SOURCE- and OBJECT-COMPUTER paragraphs.


In the data division:


WS-LIMIT is a constant, so it's good idea to declare it as such. Change its declaration to

01  WS-LIMIT CONSTANT AS 10.


or, if your compiler doesn't support that, try the common non-standard extension:

78  WS-LIMIT VALUE 10.

Some of the variables begin with WS- and some don't. I suggest getting rid of the WS- prefix: there's nothing to gain here from indicating a variable is in working-storage.
Similarly, there's nothing to gain from putting variables in the WS-STORAGE, COUNTERS and REMAINDERS group (except for the OCCURS items). 


In the procedure division:


Get rid of the redundant END-COMPUTE's. They're only needed for the SIZE ERROR handlers and clutter the code otherwise.


Quuxplusone raises a good point that FIZZ and BUZZ aren't strictly necessary. You could rewrite the EVALUATE like so, and not hide the meaning of FIZZ and BUZZ in the data division:

EVALUATE mod-three ALSO mod-five
    WHEN 0 ALSO 0
        DISPLAY "FizzBuzz"
    WHEN 0 ALSO ANY
        DISPLAY "Fizz"
    WHEN ANY ALSO 0
        DISPLAY "Buzz"
    WHEN OTHER
        DISPLAY current-num
END-EVALUATE