Why does my forest plot say a study's result is statistically significant when it's not? A large amount of computing resources is required to compute exact probabilities for the Kruskal–Wallis test. Existing software only provides exact probabilities for sample sizes less than about 30 participants. These software programs rely on asymptotic approximation for larger sample sizes.
Exact probability values for larger sample sizes are available. Spurrier (2003) published exact probability tables for samples as large as 45 participants.[7] Meyer and Seaman (2006) produced exact probability distributions for samples as large as 105 participants.[8]
 A: This discrepancy seems to come from errors in the Ozbay et al paper.
According to the Materials and Methods, the error estimates in Table 1 are standard deviations. To run the independent-samples t-test, however, you also need to take into account the number of samples; the test depends on the standard errors of the mean values, which decrease with the square root of the numbers of observations. The authors report 107 tinnitus patients and 107 matched controls.
If the error terms in Table 1 are standard deviations, as stated in their Materials and Methods, then the p-value reported by Ozbay et al is perhaps consistent with their having used the standard deviations rather than the standard errors of the means in their t-test calculations. If you take into account the 107 cases in each group, however, the standard errors would be only about 1/10 of the standard deviations (0.11 and 0.13, respectively), so that the difference of 1.59 in MPV between the 2 groups would be highly significant.
One might argue that Ozbay et al used instead a Kruskal-Wallis test,* based on the data failing to fit a normal distribution. With 107 in each group, however, the raw-observation normality assumption for the t-test is probably irrelevant. Such tests for differences between mean values depend on normality of the mean-value estimates. Thanks to the central limit theorem, those mean-value estimates with such large sample sizes are likely to have distributions close to normal. Lumley et al go through these issues in some detail, and point out the limitations and hidden assumptions in use of non-parametric tests instead. At least for variables that showed similar standard deviations between tinnitus and control groups, t-tests should give reliable results for data reported by Ozbay et al.
I can't speak to the extra heterogeneity introduced by including that paper in your meta-analysis. A quick look at it suggests that it was not the most carefully controlled study. In addition to the above apparent error in performing statistical tests, some of the underlying data are of questionable quality. For example, although the platelet counts (in units of 10^3 per mm^3) are similar between the 2 groups (225 tinnitus, 260 control) their reported standard deviations differ by more than a factor of 10 (10 and 110, respectively).
Part of doing a thorough meta-analysis is getting as much information as possible about the details of the underlying studies. In this case, I think that if you look into this paper in more detail you might be justified in discounting it based on its deficiencies.

*When comparing 2 groups as in Ozbay et al, this is the Mann-Whitney or Wilcoxon test.
A: I would like to second EdM's answer. Particularly, I would like to add some thoughts on their last comment about the paper's deficiencies. 
As EdM already stated, it is easy to confirm that the provided data and p-values do not make sense in the way that they are reported by Ozbay et al. Maybe the data is actually extremely far from normality, and then a Mann-Whitney-Test, which does not care about the means, could provide the p-values reported. But this is unlikely, to say the least. Or maybe they actually run a more complex model, e.g. a multivariable regression, and all these p-values are adjusted for covariables. But in their Methods, they say otherwise. 
Ozbay et al. do not report any power analysis, and therefore any conclusions like "the parameters were similar" based on p-values are invalid. We need at least a confidence interval and enough statistical power to conclude something about similarity. 
The description of the statistical analysis is neither insightful nor does it make sense. A Kruskal-Wallis test for two groups is actually a Mann-Whitney test (or Wilcoxon test). Reporting the mean when doing a Mann-Whitney test is not helpful. They did not adjust for any potential confounders.
I am definitely no expert in this field, but I would question whether matching just based on age and sex is appropriate. Patients with Tinnitus have higher stress levels, and the association between stress level and neutrophil to lymphocyte ratio is well known. It is not surprising that Tinnitus patients have higher NLR than healthy controls. (well, actually in their figure, the two groups look quite similar. I would not even trust that p<0.05)
You say it is a highly cited paper, but my naive google scholar request revealed just 17 citations in the past 5 years, and most of them seem to be from Turkey, too. This is not neccesarily a bad sign, of course, but I am always a bit sceptical, if a study is cited mainly by people from the same regions. Sometimes this is an indicator for self-citing circles instead of good quality of the research.
However, the best way to get clarity is to ask the authors for the raw data. Maybe there is a good explanation for all of this. Maybe it were just simple mistakes. And if they do not provide you with the raw data, well, then you might put two and two together and maybe not fully trust the results.
