Irritability is a pretty broad concept. Various aspects/measures of it have shown various degrees of heritability. For example, just in one twin study (Coccaro, 1997):
To determine the degree of genetic and environmental influences on assessments of aggression and irritability in male subjects, the "Motor Aggression" subscales of the Buss-Durkee Hostility Inventory (BDHI) were mailed to 1208 male twins in the Vietnam Era Twin Registry. Data from monozygotic 182 and 118 dizygotic twin pairs were available and were analyzed using model-fitting procedures. Three of the four BDHI subscales demonstrated significant heritability of a nonadditive nature: 40% for Indirect Assault, 37% for Irritability, and 28% for Verbal Assault. Additive genetic variance accounted for 47% of the individual differences for Direct Assault. Nonshared, but not shared, environmental influences contributed to explaining the variance in the model, with estimates ranging from 53% (Direct Assault) to 72% (Verbal Assault). Because some of these BDHI scales have been shown to correlate with indices of central serotonin function, it is possible that impulsive aggression, as reflected by these scales, is heritable in men.
And a study from the genomic era (Riglin, 2017), alas only on irritability in children:
Severe irritability is one of the commonest reasons prompting referral to mental health services. It is frequently seen in neurodevelopmental disorders that manifest early in development, especially attention-deficit/hyperactivity disorder (ADHD). However, irritability can also be conceptualized as a mood problem because of its links with anxiety/depressive disorders; notably DSM-5 currently classifies severe, childhood-onset irritability as a mood disorder. Investigations into the genetic nature of irritability are lacking although twin studies suggest it shares genetic risks with both ADHD and depression. We investigated the genetic underpinnings of irritability using a molecular genetic approach, testing the hypothesis that early irritability (in childhood/adolescence) is associated with genetic risk for ADHD, as indexed by polygenic risk scores (PRS). As a secondary aim we investigated associations between irritability and PRS for major depressive disorder (MDD). Three UK samples were utilized: two longitudinal population-based cohorts with irritability data from childhood (7 years) to adolescence (15–16 years), and one ADHD patient sample (6–18 years). Irritability was defined using parent reports. PRS were derived from large genome-wide association meta-analyses. We observed associations between ADHD PRS and early irritability in our clinical ADHD sample and one of the population samples. This suggests that early irritability traits share genetic risk with ADHD in the general population and are a marker of higher genetic loading in individuals with an ADHD diagnosis. Associations with MDD PRS were not observed. This suggests that early-onset irritability could be conceptualized as a neurodevelopmental difficulty, behaving more like disorders such as ADHD than mood disorders.
Note that since polygenic risk scores currently only cover a small fraction of the heritability found in twin studies (for almost any psychiatric/psychological issue/construct), genomic results are rather tentative. Usually the presence of a gene overlap means something, but the absence of a genetic overlap can simply mean there hasn't been enough data.
And Riglin point out to a recent review Vidal-Ribas (2016) of which they are aware they cannot explain genomically. The latter found/concluded that
irritability forms a distinct dimension with substantial stability across time, and that it is specifically associated with depression and anxiety in longitudinal studies. Evidence from genetic studies reveals that irritability is moderately heritable, and its overlap with depression is explained mainly by genetic factors. [...]
The genetic contribution to the variation of irritability is approximately 30% to 40% in both adults and adolescents. This is close to the heritability estimates for depression and anxiety. Genetic influences on irritability increase slightly over time in males and decrease in females, and unique (as opposed to shared) environmental factors explain most of the remaining (non-genetic) variance in both.
I highly suggest reading this review if you're unfamiliar with the psychiatric approach to irritability because it has a good, detailed background on that (including the substantial changes in DSM-5, which were applied only to children though.) And since the focus on irritability as separate entity is pretty new in psychiatry, the authors of this review note for instance that
Probably the main limitation in the study and treatment of irritability so far is the lack of high-quality measures. Most of the measurement of irritability has been done with instruments created ad hoc, that is, through extracting items from existing scales or interviews that were not intended primarily to measure irritability.
For instance, the heritability data on adults they cited from Coccaro (1997) with which I started this answer, whereas for adolescents (as you probably guessed by now), there are newer studies available; the data cited on the latter in the review is from Stringaris (2012) and there's more specific longitudinal data in Savage (2015).