Will Sirius B start accreting from A and become a supernova type Ia?
Sirius B is a massive white dwarf of 1 Solar mass, orbiting at about 25 AU distance from the 2 Solar mass Sirius A. As it evolves and expands, will the A star start shedding matter to the white dwarf, and when will this start happening? Will the Sun be at a safe distance when/if it does happen, or is Sirius our doom?
Will Sirius B start accreting? Yes, it is doing so now. Sirius A will have a wind and some of that wind will be captured by the white dwarf.
The effectiveness of wind capture is a strong function of relative wind speed. An analytic approximation to the accretion rate, known as Bondi-Hoyle accretion, goes as the inverse cube of the relative speed. In its present evolutionary state, the mass loss from Sirius A will be relatively weak (like the Sun) and relatively fast (like the Sun). This disfavours any significant accretion by the white dwarf.
However, in the later stages of its life, Sirius A will swell to become an asymptotic giant star. The envelopes of such stars are gradually (on timescales of millions of years) blown away quite slowly by a dusty wind. If Sirius A is around 2 solar masses now, it will lose about 1.4 solar masses during this phase at speeds of only 10-20 km/s.
Only a fraction of this mass may be accreted by the white dwarf, because the separation between the stars is still large at 25 AU (and will become larger as mass is lost from the system) compared with the likely terminal size of Sirius A (probably of order 2 au). If you look at the likely Roche lobe size, then Roche lobe overflow would require A to reach about 40% of the separation, which isn't going to happen. Exactly what fraction is captured by the less efficient wind accretion process (the majority will likely disappear into space and widen the orbit) depends strongly on the wind speed, which is hard to predict.
Even if Sirius B could accrete the 0.35-0.4 solar masses (I think that is unlikely, but lack the wherewithal to do the hydrodynamic simulation) it needs to become unstable, it is not clear if that mass will "stick". A build up of hydrogen-rich material can ignite and explode in a nova (not supernova) on the surface of a white dwarf, causing mass loss!
Finally, when will this happen? Well Sirius is probably about 300 million years old now and has another perhaps 500 million years before it starts to evolve in the way I described. It will be nowhere near the Sun then.