This is a fairly open-ended question... One example of the potentially negative consequences of overexpressed BDNF is its effect on the likelihood and frequency of epileptic seizures.
I guess the short version of this effect is that BDNF appears to promote neuronal growth, including neurogenesis, axonal and synaptic sprouting, and neuronal excitation, in exactly the areas of the brain that are associated with epilepsy. Epileptic seizures are essentially hyper-excitations of neurons, so any factor that increases the excitability of neurons in brain areas associated with seizures can be a culprit.
A good, fairly recent, and open access review of the role of BDNF and its primary receptor TrkB in epilepsy is McNamara & Scharfman (2012):
Intrahippocampal infusion of BDNF and transgenic overexpression of
BDNF or TrkB increase seizure susceptibility or severity. Conditional
knockout of TrkB eliminated epileptogenesis altogether in the kindling
That is, in epileptic mice, adding BDNF increased seizure frequency, and removing it eliminated seizures entirely. The authors suggest several mechanisms by which this might happen:
... long-term potentiation (LTP) of excitatory synapses between
principal cells may contribute to limbic epileptogenesis; that
is, potentiation of these synapses may facilitate propagation of
seizure activity through synaptically-coupled neuronal populations
throughout the limbic system and beyond.
One interesting possibility is that enhanced TrkB activation reduces
expression of the K-Cl cotransporter, KCC2, ... an important molecular
and cellular mechanism contributing to limbic epilepsy.
The effect of estrogen on NPY levels could be mediated by BDNF,
because BDNF induces NPY synthesis following TrkB activation
In other words, (1) BDNF-induced LTP - the mechanism that underlies memory and learning - increases the excitability of neurons, potentially causing seizures; (2) BDNF may inhibit KCC2, itself a synpatic inhibitor, again increasing the excitability of neurons; and (3) hormone-induced BDNF may moderate neuron hyper-excitability through its effect on yet another protein, the neurotransmitter NPY.
Another recent review, though not open access, is Iughetti et al (2018). Similar to the previous review:
... BDNF has excitatory effects in neuronal cultures and animal brain
slices. Furthermore, both BDNF and its conjugated receptor (... TrkB)
are increased in animal models and humans with epilepsy, particularly
in the temporal and hippocampal areas. Acute injection of BDNF in
brain of mice induces seizures, which are almost or totally abolished
[by] blocking its transcription and pathway.
The authors also implicate the effects of BDNF on axonal sprouting and neuronal hyper-excitability, KCC2 and NKCC1 downregulation, and NPY synthesis on seizure likelihood and frequency.
Neurochemistry is complex and not well understood, and there is indication that BDNF can also have protective properties against epilepsy under certain conditions (eg, long-term in low doses). Lots more detail is available in the reviews cited.