You should first parse the strings to TimeSpan
TimeSpan start = TimeSpan.ParseExact(startTime, "hhmm", CultureInfo.InvariantCulture);
TimeSpan close = TimeSpan.ParseExact(closeTime, "hhmm", CultureInfo.InvariantCulture);
then you can use this LINQ query:
List<TimesInfo> inRange = list
.Where(ti => ti.startTime.TimeOfDay >= start && ti.closeTime.TimeOfDay <= close)
.ToList();