In What happened to disease in D&D 5e? we see that disease may not be a specific mechanic. However, it seems to be conflated quite a bit with poison, for example:
Here's an example of an actual disease from another disease riddled classic the Otyugh (from the HOTDQ supplement)
If the target is a creature, it must succeed on a DC 15 Constitution saving throw against disease or become poisoned until the disease is cured.
In this case, is the saving throw "against disease" or "against poison" (for the purposes of the poison mechanic vis-a-vis the Dwarven Poison Resistance). I think from reading this post that some may say that the "against poison" of the Dwarven Resilience is specifically against the poisoned condition.
Likewise, Bottle Fever in the SRD says:
A creature who drinks rancid alcohol must succeed on a DC 12 Constitution saving throw or contract bottle fever. The first symptoms of bottle fever begin to show in 1d10 hours. An infected creature gives off a strong alcoholic odor. When the creature performs an act requiring physical exertion, like climbing a wall or fighting, it must succeed on a DC 10 Constitution saving throw or become poisoned for 1 hour. If the creature succeeds on the saving throw it does not need to make another saving throw for this effect for 1 hour.
If the disease gives you the 'poisoned' condition such as in the case of the Otyyugh does Dwarven Resilience apply?
If the disease gives you the 'poisoned' condition later, such as in Bottle Fever where one would have to fail two saves — does Dwarven Resilience apply to the first, second, both or neither saves?
A save vs disease is against disease
If a disease causes the poisoned condition, that is merely a mechanical symptom of the disease. If the description states that the saving throw is made against disease, Dwarven Resilience won't help as it grants advantage specifically to saving throws against poison. If the disease happened to cause poison damage, however, Dwarven Resilience would give resistance to the damage.
A creature that is immune to the poisoned condition will not be affected by the disease, whether it makes a save or not. Similarly, a creature that is immune to disease won't be affected either.
The conflation is compounded by spells such as Contagion, which requires the target to be affected by the poisoned condition before any of the nasty diseases of that spell can take effect. Ultimately, you will need to read the description of a disease carefully to determine whether it is really a poison or simply causes the poisoned condition.