As Wikipedia explains:
Heritability is a statistic used in the fields of breeding and genetics that estimates the degree of variation in a phenotypic trait in a population that is due to genetic variation between individuals in that population. In other words, the concept of heritability can alternately be expressed in the form of the following question: "What is the proportion of the variation in a given trait within a population that is not explained by the environment or random chance?"
Naively, negative heritability shouldn't exist if the assumptions of the probability model used to estimate heritability are true. But, sometimes experimental data appears to imply that something has negative heritability.
Actually, the concept of a negative probability isn't entirely foreign to mathematics and science. In quantum mechanics, you can have negative probabilities in linear contributions to the probability of an observable so long as all observables have non-negative probabilities. Something similar could be at work in the area of gene expression.
So, it isn't unthinkable that negative heritability does exist. A new pre-print considers this issue:
We consider the problem of interpreting negative maximum likelihood estimates of heritability that sometimes arise from popular statistical models of additive genetic variation. These may result from random noise acting on estimates of genuinely positive heritability, but we argue that they may also arise from misspecification of the standard additive mechanism that is supposed to justify the statistical procedure. Researchers should be open to the possibility that negative heritability estimates could reflect a real physical feature of the biological process from which the data were sampled.
David Steinsaltz, Andy Dahl, Kenneth W. Wachter "On negative heritability and negative estimates of heritability" bioRxiv (May 14, 2018).