The Boost C++ libraries include an implementation of Fibonacci heaps in boost/pending/fibonacci_heap.hpp. This file has apparently been in pending/ for years and by my projections will never be accepted. Also, there have been bugs in that implementation, which were fixed by my acquaintance and all-around cool guy Aaron Windsor. Unfortunately, most of the versions of that file that I could find online (and the one in Ubuntu's libboost-dev package) still had the bugs; I had to pull a clean version from the Subversion repository.
I was able to compile dijkstra_heap_performance.cpp against a modified version of dijkstra_shortest_paths.hpp to compare Fibonacci heaps and binary heaps. (In the line typedef relaxed_heap<Vertex, IndirectCmp, IndexMap> MutableQueue, change relaxed to fibonacci.) I first forgot to compile with optimizations, in which case Fibonacci and binary heaps perform about the same, with Fibonacci heaps usually outperforming by an insignificant amount. After I compiled with very strong optimizations, binary heaps got an enormous boost. In my tests, Fibonacci heaps only outperformed binary heaps when the graph was incredibly large and dense, e.g.:
Generating graph...10000 vertices, 20000000 edges.
Running Dijkstra's with binary heap...1.46 seconds.
Running Dijkstra's with Fibonacci heap...1.31 seconds.
Speedup = 1.1145.
As far as I understand, this touches at the fundamental differences between Fibonacci heaps and binary heaps. The only real theoretical difference between the two data structures is that Fibonacci heaps support decrease-key in (amortized) constant time. On the other hand, binary heaps get a great deal of performance from their implementation as an array; using an explicit pointer structure means Fibonacci heaps suffer a huge performance hit.
Therefore, to benefit from Fibonacci heaps in practice, you have to use them in an application where decrease_keys are incredibly frequent. In terms of Dijkstra, this means that the underlying graph is dense. Some applications could be intrinsically decrease_key-intense; I wanted to try the Nagomochi-Ibaraki minimum-cut algorithm because apparently it generates lots of decrease_keys, but it was too much effort to get a timing comparison working.
Warning: I may have done something wrong. You may wish to try reproducing these results yourself.
Theoretical note: The improved performance of Fibonacci heaps on decrease_key is important for theoretical applications, such as Dijkstra's runtime. Fibonacci heaps also outperform binary heaps on insertion and merging (both amortized constant-time for Fibonacci heaps). Insertion is essentially irrelevant, because it doesn't affect Dijkstra's runtime, and it's fairly easy to modify binary heaps to also have insert in amortized constant time. Merge in constant time is fantastic, but not relevant to this application.
Personal note: A friend of mine and I once wrote a paper explaining a new priority queue, which attempted to replicate the (theoretical) running time of Fibonacci heaps without their complexity. The paper was never published, but my coauthor did implement binary heaps, Fibonacci heaps, and our own priority queue to compare the data structures. The graphs of the experimental results indicate that Fibonacci heaps slightly out-performed binary heaps in terms of total comparisons, suggesting that Fibonacci heaps would perform better in a situation where comparison cost exceeds overhead. Unfortunately, I do not have the code available, and presumably in your situation comparison is cheap, so these comments are relevant but not directly applicable.
Incidentally, I highly recommend trying to match the runtime of Fibonacci heaps with your own data structure. I found that I simply reinvented Fibonacci heaps myself. Before I thought that all of the complexities of Fibonacci heaps were some random ideas, but afterward I realized that they were all natural and fairly forced.