You could turn off http request logging in /etc/influxdb/influxdb.conf and reload InfluxDB:
[http]
log-enabled = false
You could also look into changing your log rotation settings, this is system dependent but on debian you may just need to modify /etc/logrotate.d and check man logrotate.
Is it still the logs taking all the space, remember the database also uses /var so at some point I think you will just need more space.
I know you could completely disable logging, at least when using sysvinit, by setting STDERR=/dev/null, here are the logging docs. Of course this will make it very hard to debug what is going on so I don’t really recommend it.