I'll start by saying that any veteran who dies as a result of their military service--so if they die as a result of an illness or injury for which they have a disability pension or disability award--Veterans Affairs will provide funeral and burial assistance without regard to their financial means at the time of their death.
In addition, Second World War and Korean War veterans who have insufficient means.... And that's based on a means test that takes into consideration whether or not they have a surviving dependant, usually a spouse, and it looks at what assets there are between the veteran and the surviving spouse. A couple of things are exempted--the principal residence and one automobile. Other assets are considered as part of the means test, the asset test, to determine whether or not they are eligible. If they have more than $12,700 in those other more liquid assets, they are expected to contribute to the cost of the funeral and burial. If they are a few hundred dollars over that line, then that is what their contribution would be, a few hundred dollars, and they would receive a partial funeral and burial grant.
For post-Korean War veterans, there is one difference in the eligibility criteria. In order to be considered for the means test, they must be in receipt of a departmental benefit. That could be a disability pension, a disability award, an income support, or, right now, earnings-loss compensation. If they die while in receipt of any of those benefits, then exactly the same means test is applied.