How does the law of diminishing marginal utility affect the demand for a Veblen good?

By definition, a Veblen good is one for which the quantity demanded increases as the price increases, which is an apparent contradiction of the law of demand. Examples of Veblen goods are normally very expensive products, such as designer jewellery, branded watches and luxury cars. It is an exception to the normal rules regarding the relationship between price and demand.
The purchase of more expensive goods confers status on the purchaser - a process which Veblen called conspicuous consumption.
In answer to your question, the law of diminishing marginal utility states that as more units of a particular good is consumed, the consumer gains less utility from it. It explains the downward slope of the demand curve. This is where the contradiction lies, as for Veblen goods, the curve slopes upwards.
The higher price of a Veblen good does not directly increase the desirability of possessing the good, but rather is being used as a signal of other information.
So, the law of diminishing marginal utility arguably still holds. Initially, marginal utility increases, until it reaches it’s maximum. After that, an extra unit of consumption of a Veblen good declines marginal utility.