Usually such a large theta reflects that there is simply no overdispersion in the data, possibly even some underdispersion.
As a simple example consider the following artificial Poisson-distributed regression model:
d <- data.frame(x = runif(500))
d$y <- rpois(500, exp(0 + 2 * d$x))
Thus, this is a plain Poisson GLM with intercept 0 and slope 2. Of course, these coefficients could also be recovered from a negative binomial hurdle, just by estimating more coefficients than necessary (count theta plus zero hurdle coefficients).
m1 <- hurdle(y ~ x, data = d, dist = "negbin")
coef(m1, model = "count")
## (Intercept) x
## -0.01399583 2.06624937
Thus, the coefficients are very close to their true values and theta = 69.3 is already rather close to infinity (= Poisson distribution).
If we now make the data somewhat more underdispersed by cutting only very few extreme counts, theta can go up extremely: First, we just omit a single observation, the one with the maximum y = 17:
hurdle(y ~ x, data = d, dist = "negbin", subset = y < max(y))$theta
Omitting all y > 10 (i.e., nine observations), theta becomes even more extreme:
hurdle(y ~ x, data = d, dist = "negbin", subset = y <= 10)$theta
Thus, the extreme theta reflects that you don't need a negative binomial model here. Possibly a Poisson distribution fits well enough, if the underdispersion is not too pronounced. A useful graphical check whether the Poisson distribution is an ok fit for your data is the rootogram, see Kleiber & Zeileis (2016, The American Statistician, doi:10.1080/00031305.2016.1173590).