First, getting into really major concerns, this paper suffers from some serious potential for omitted variable bias. The goal of this study is to find the causal effect or treatment effect of Airbnb on violent crime: "This study tested the hypothesis that the arrival and growth of Airbnb, or home-sharing platforms in general, may increase crime and disorder in neighborhoods..." In other words, we want to know what would happen to violence if we could push a lever and increase or decrease Airbnb directly. A regression of violence on Airbnb tells us how violence tends to change with changes in Airbnb in the data. If there's some other factor that affects both Airbnb and violence (or affects one and is affected by the other), then the regression is going to find the mismash of the causal effect of Airbnb and the causal effect of that other factor and put it all on Airbnb.

This is especially problematic because the authors find changes in Airbnb penetration are correlated with census tract characteristics:

For Airbnb density (Fig 3a), we see that census tracts in the urban center (northeast on the map) show relatively high Airbnb presence from the beginning, but that in recent years the tracts with the highest level of Airbnb penetration emanate further out into surrounding, more residential neighborhoods.

This can be addressed by controlling for the confounding factor. Alternatively, we can look at a specific scenario where Airbnb penetration changed for reasons that almost certainly couldn't affect violence, and use that variation to find out how violence responds to Airbnb penetration.

Unfortunately, the paper only controlled for median income in the primary specification and percentage of black/Hispanic residents and homeownership rates in the robustness check (these additional controls are not in the database). This paper simply does not do enough to eliminate potential other explanations for why Airbnb penetration may be correlated with violence in a census tract without causing it.

I'm not that familiar with the crime literature, but from a very brief look at the abstract of one random paper, factors like economic inequality, poverty rates, population density, and divorce rates could predict violent crime. All of these could affect Airbnb penetration, such as through property prices (another potential confounding variable) or through amenities for tourists.

For example, increases in population density could both increase crime and increase Airbnb penetration (more amenities or maybe it's easier to buy new houses than existing houses). Or maybe Airbnb penetration increases more in areas with higher income inequality because there are cheap units near areas with rich amenities, and it's higher income inequality driving crime and Airbnb penetration. It could even be something unrelated to Airbnb that just so happened to affect urban and residential areas differently over these years. For example, maybe these regions had different changes in police presence during that time.

Second, another major concern is the potential nonstationarity in violent crime and Airbnb penetration. Airbnb has a very clear trend of increasing over time, as shown in Figure 1 of the paper. You can imagine violent crime also having a very clear time trend where this year's value depends heavily on last year's value. Controlling for year and neighborhood fixed effects, both a regression of violence and Airbnb penetration on its lagged (prior-year) value finds a coefficient of roughly 1, which indicates the presence of a time trend (technically you need a coefficient greater than or equal to 1, but coefficients close to 1 also produce similar problems with short time series like this one).

When you regress any non-stationary data series on another, you will always find a strong relationship between the two. That's true even with non-sensical relationships like GDP in the US and total recorded rainfall in Cambodia since Jan 1, 1900. You can think of time as the confounder which affects both series. Here, time is increasing both Airbnb penetration and violence. Even if they have nothing to do with each other, a regression will find a strong correlation between the two.

To avoid spurious correlations due to time trends, the most common way is to remove the time trend by looking at the change in the variables rather than their absolute values. That cancels out the time trend and allows for proper inference. This is why studies in finance usually look at returns (changes in values) rather than asset values. If we do that for this data, we again find the coefficient on Airbnb penetration becomes insignificant.
To conclude, the very least we can say about the results of this paper is that it is incomplete. Plenty of standard controls for crime are not included, which weakens the ability of the paper to argue Airbnb causally leads to more violence. Further, the authors did not properly handle the presence of non-stationary data.