There are a number of cases where the existence of large cardinals (or even their consistency) has "positive" consequences, e.g. determinacy hypotheses or uniformity of abstract elementary classes and accessible categories.

However, one phenomenon stands out as being the opposite: several nice properties in analysis are only closed under products if the cardinality of the index set is below the least measurable cardinal. The simplest manifestation of this is probably that a discrete space $X$ is realcompact (i.e. embeds into a power of $\mathbb{R}$) iff there are no measurable cardinals or $|X|$ is less than the least measurable cardinal. This is very unintuitive to analysts, and occasionally published papers make the mistake of assuming this phenomenon does not exist (or know that it exists and quote it incorrectly because of changing terminology related to measurability over time).

Are there any other examples of consequences of large cardinals that stand out as being unintuitive and somewhat "negative" for ordinary mathematics?