I am a holder of a Venezuelan passport, and I am a permanent resident in the United States. However, my passport is an older version without the biometric strip, but it does not expire for another two years, and I would like to travel to Europe as a visitor for a week, or maybe less. If I am traveling to any country in Europe for less than 90 days with my Venezuelan passport, do I need to apply for a Schengen visa?
As far as I know there is no general requirement to hold a machine-readable or biometric passport to visit the Schengen area without visa (unlike the US Visa Waiver Program). Except nationals of a handful of countries like Albania or Bosnia-Herzegovina (who do require a biometric passport), visitors are exempted from the visa requirement (or not) based on citizenship. As a Venezuelan citizen, you can visit those countries without visa. Romania, Bulgaria, Croatia, and Cyprus also apply the same rules.
I don't think the UK has a biometric passport requirement either but Venezuelan citizens need a visa in any case so you won't be able to go there.
I don't know about other European countries.