Welcome to TFF.
I agree with the above. Neither species will live very long in either tank for a couple of reasons.
First, Tiger Barbs. This fish is a notorious fin nipper, and it must be maintained in a group of 8 minimum but 10-12 would be better. However, this requires a 30 gallon tank (base dimensions of 30 by 12 inches (75 by 30 cm) for just this fish group. In smaller tanks or smaller numbers, the fish will soon be severely stressed, which means further problems and a short lifespan because fish simply cannot cope with such things. Returning these fish is highly advisable.
Second, the neon tetra. For the best health, this fish should have at minimum a 20 gallon tank with base dimensions of 24 by 12 inches (60 by 30 cm) and be in a group of no fewer than six but a few more will be much better. All nine neons in the 10g on their own could work, though long term not in the best interests of the fish, so I would suggest returning these too.
There are fish suited to a 10 gallon tank, and we can go into the suitable species once we have the water parameters confirmed; the hardness (GH) is critical, and then pH [8.2 is given here, but without knowing the GH we cannot know if this may lower or not, and neons are soft water fish]. You should be able to ascertain the GH (general or total hardness) from your water authority, check their website.
GH is important because it involves the dissolved mineral, primarily calcium and magnesium, in the water. Whatever comes out of the tap willnot change much if at all in an aquarium. The pH is tied to the GH but it might lower. Each freshwater fish species has evolveed to function best in very specific parameters. There are basically soft water species and hard water species, and some that manage well somewhere in the middle. So GH is very important to fish.
Concerning the rapid deaths, this could be cycling-related or something else. I never guess with disease-type issues, but the above is going to cause trouble going forward.