I swear, some superstitions just won't die. No, comet ISON's dust is not going to infect us with "the black death" better known as Yersinia pestis. Few organisms can survive the harsh conditions of outer space. Some extremophiles can, but the most "space worthy" organisms on earth are not sources of plague. Some of the largest sources for potential interplanetary contamination by our own probes traveling to places like the moon, for example, are about organisms hidden inside the spacecraft or in compartments that are shielded from the sun's unfiltered UV light.
[link to ntrs.nasa.gov]
Yersinia Pestis, the cause of the black death ( [link to www.ncbi.nlm.nih.gov] [link to www.nature.com] ), is particularly susceptible to UV light like its other gram negative brethren.
[link to aem.asm.org]
It wouldn't stand a chance floating along attached to a cometary dust particle in space. And yes, the dust that reaches us from ISON (if any at all - the thing is under-performing expectations) will be quite small.

[link to www2.ess.ucla.edu]

As you can see, if you solve for dust grain size, it is inversely proportional to the acceleration it experiences relative to the comet. The reason is simple; the dust tail is formed by solar radiation pressure, which has a greater influence over smaller particles. If a particle were to have left ISON on January 1, 2012, well before it was even discovered, and at a sun distance in excess of 8.6 AUs, it would need a total delta-V (change in velocity) of about 2265 m/s to reach earth on January 12, 2014, when we are potentially going to encounter some of ISON's dust.

[link to img837.imageshack.us]
(I manually set the virtual spacecraft in the sim to the exact same state vectors as ISON for that time, then saw how much delta-V was needed to reach earth on January 12, 2014 - the number is listed under dV applied as Tot and is in m/s - I tested it to make sure the course actually did bring the test particle to earth on January 12 2014)

That's 64195200 seconds later. For the sake of simplicity, let's say that the acceleration needed (alpha in the above formula) is simply the delta-V divided by the time to reach earth (it's really more than that - the closer it gets to earth, the more delta-V is needed to adjust the course to reach earth from the course the nucleus is on, so we're being generous to the final grain size in this assumption). That works out to an alpha of 3.528x10^-5 m/s^2. Solving for the dust grain size, that works out to about 10 microns. That is about the size of the dust grains that can therefore be expected to reach us on January 12th, provided there are enough particles of that size and accelerated by the sun's light along the right velocity vector to ultimately reach us.