It has already been pointed out in the comments that there are ways to get a CPU profile from the VM on Linux (https://code.google.com/p/dart/wiki/Profiling).
As far as I understand what you are really looking for is to get a heap or memory profile. While it is possible to print an object histogram when the program terminates (see below), we do not have any convenient way to get the object histogram while your server is running. We do hope to be able to add this capability over the next months.
To print the object histogram when the Dart script exits, you should pass the flag --print_object_histogram to the Dart VM. This will print the averages of the live objects at the end of each major GC over the life of the program. This can be fine to get a quick overview, but is not ideal to track down and identify real problems.