What is the significance of Dick Van Dyke playing both Mr Dawes and Bert in Mary Poppins? I have a feeling that something of this sort might have been intended, but is there any official word on such? Has there been other speculation?
There isn't any behind the scenes meaning of why Van Dyke played both roles. Van Dyke had lobbied Walt Disney for the part for some time before Disney acquiesced to allow him to play it. According to this article from the LA Times:
Van Dyke had to cajole Walt Disney into giving him the part because Disney thought Van Dyke -- then in his 30s -- was too young to be the ancient moneyman. The actor reportedly won him over by acing a screen test, agreeing to portray Dawes for free and making a donation to the California Institute of the Arts, which Disney co-founded.
NOTE: Emphasis mine.
It seems it was just a matter of Van Dyke wanting the part and being a "go-getter" was able to convince Disney in the end.