If you insist on observing the exploding Betelgeuse at peak brightness, you could potentially damage your eye. The complete answer enters the realm of physiology. Here I'll discuss the astronomical parts:
Betelgeuse will explode as a type II supernova, the typical brightness of which is around $M \sim -17$. With a distance of $d\simeq200\,\mathrm{pc}$, its distance modulus is
$$
\mu = 5\log(d/\mathrm{pc}) - 5 \simeq 6.5,
$$
so its apparent magnitude will be
$$
m = M + \mu \simeq -10.5.
$$
For these calculations I assume that the Sun is the threshold for damaging your eye (a brief look at the Sun is okay, a longer look will cause permanent damage. But… physiology…). The Sun has an apparent magnitude of $m_\odot = -26.7$, i.e. it is $\Delta m = 16.2$ magnitudes brighter. In other words, Betelgeuse will be
$$
f = 10^{\Delta m/2.5} \simeq 3\times10^6
$$
times dimmer than the Sun.
However, the Sun is an extended source, spanning an angle of roughly $\theta_\mathrm{Sun} = 32$ arcminutes across. In contrast, Betelgeuse is a point source, which when transferred through the atmosphere and the telescope, is spread out over $\theta_\mathrm{Bet} \sim$ a few square arcseconds. Thus its light will be more concentrated; i.e. it will be much brighter, but it will hit a much smaller area of your retina. However, your eye will also move around, smearing out the light. Not being a physiologist, for the sake of this calculation I assume that the light is smeared out over a disk 1 arcminute across (about the size of a planet seen from Earth).
Thus, the factor $f$ will itself be a factor $(\theta_\mathrm{Sun} / \theta_\mathrm{Bet})^2 \simeq 1000$ times larger — that is, Betelgeuse is only $\sim 3\,000$ times dimmer than the Sun.
Hence, for our assumptions your eye will be damaged if you observe exploding Betelgeuse through a telescope with an area $\sim 3\,000$ larger — or roughly 55 times wider — than your pupil. In bright light, the pupil contracts to roughly 3 mm in diameter, so if observing through a telescope of 16 cm or larger, you could damage your eye.
Based on evolutionary models of Betelgeuse, Dolan et al. (2016) estimate an apparent magnitude of $m=-12.4$, i.e. roughly 6 times brighter than our estimate. This would mean that you only need a 7 cm telescope to damage your eye.
However, as Mark writes in his answer, supernovae don't increase to their peak brightness in matters of seconds, but rather in matters days (roughly half a mag per day), so you have plenty of time to look away.
