  Detections of gravitational waves emitted from binary black hole coalescences
allow us to probe the strong-field dynamics of general relativity (GR). One can
compare the observed gravitational-wave signals with theoretical waveform
models to constrain possible deviations from GR. Any physics that is not
included in these waveform models might show up as apparent GR deviations. The
waveform models used in current tests of GR describe binaries on quasicircular
orbits, since most of the binaries detected by ground-based gravitational-wave
detectors are expected to have negligible eccentricities. Thus, a signal from
an eccentric binary in GR is likely to show up as a deviation from GR in the
current implementation of these tests. We study the response of four standard
tests of GR to eccentric binary black hole signals with the forecast O4
sensitivity of the LIGO-Virgo network. Specifically, we consider two
parameterized tests (TIGER and FTI), the modified dispersion relation test, and
the inspiral-merger-ringdown consistency test. To model eccentric signals, we
use non-spinning numerical relativity simulations from the SXS catalog with
three mass ratios $(1,2,3)$, which we scale to a redshifted total mass of
$80M_\odot$ and luminosity distance of $400$ Mpc. For each of these mass
ratios, we consider signals with eccentricities of $\sim0.05$ and $\sim 0.1$ at
$17$ Hz. We find that signals with larger eccentricity lead to very significant
false GR deviations in most tests while signals having smaller eccentricity
lead to significant deviations in some tests. For the larger eccentricity
cases, one would even get a deviation from GR with TIGER at $\sim 90\%$
credibility at a distance of $\gtrsim 1.5$ Gpc. Thus, it will be necessary to
exclude the possibility of an eccentric binary in order to make any claim about
detecting a deviation from GR.
