Generally speaking, the more popular or useful an ancient/medieval text, the more likely it is to receive not just one but successive "modern" translations. The most prominent example is certainly the alphabet soup that is English translations of the Christian Bible--KJV, NIV, RSV, NRSV, ESV, NLT, etc. You can also see the principle in action on the Internet, especially sites like archive.org. Many translations from the 19th century are out of copyright and available online, and often they're different from a translation currently in publication of the same text.

Change in individual word meaning of the modern word would of course be reflected in a translation--I doubt any 21st century translator would use "gay" to mean happy--but usually a new translation comes about for more complex reasons. The big one related to modern language is actually style. When you read a 19th century translation, usually you know it's a 19th century translation instead of a more recent one. Also, translation is an art rather than a science. Trends and even individual translators change over time in whether they prefer to capture the word-for-word of a text as closely as possible, or capture the meaning. If you are a fan of musical theatre and listen to your favorite shows in other languages, you'll generally come across the second.

The other big category of change relates to changing interpretation of the text. Languages are not cyphers; they do not map word-for-word onto each other. One Latin word can mean six different English words depending on context, at least two of which are generally opposites of each other but most of which are closely related just with slightly different connotations. Over time, different choices can add up to different shades of meaning. However, if you're asking "will an English word be invented that captures the specific meaning-range of the Latin word," no, language doesn't really work that way.

The real fun here is of course the Bible. There's a reason many Calvinist-leaning churches use the ESV, scholars prefer the NRSV, and so forth. Different word choices tilt meanings.